Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 946491 10 pageshttpdxdoiorg1011552013946491

Research ArticleFinite-Time Stability Analysis and Control for a Class ofStochastic Singular Biological Economic Systems Based onT-S Fuzzy Model

Shuangyun Xing12 Qingling Zhang1 and Yi Zhang3

1 Institute of Systems Science Northeastern University Shenyang Liaoning 110189 China2 College of Science Shenyang Jianzhu University Shenyang Liaoning 110168 China3 School of Science Shenyang University of Technology Shenyang Liaoning 110870 China

Correspondence should be addressed to Qingling Zhang qlzhangmailneueducn

Received 28 February 2013 Accepted 23 May 2013

Academic Editor Qun Lin

Copyright copy 2013 Shuangyun Xing et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper studies the problem of finite-time stability and control for a class of stochastic singular biological economic systems Itshows that such systems exhibit the distinct dynamic behavior when the economic profit is a variable rather than a constant Firstlythe stochastic singular biological economic systems are established as fuzzy models based on T-S fuzzy control approach Thesemodels are described by stochastic singular T-S fuzzy systemsThen novel sufficient conditions of finite-time stability are obtainedfor the stochastic singular biological economic systems and the state feedback controller is designed so that the population (stateof the systems) can be driven to the bounded range by the management of the open resource Finally by using Matlab softwarenumerical examples are given to illustrate the effectiveness of the obtained results

1 Introduction

At present mankind is facing the problems of shortage ofresources and worsening environment So there has beenrapidly growing interest in the analysis and modeling ofbiological systems From the view of human needs theexploitation of biological resources and harvest of populationare commonly practiced in the fields of fishery wildlife andforestry management which has already been tackled inthe classical literature (see [1ndash5]) In these works the mainobjective was to maximize a utility function representingthe sustainable economic rent (ie net economic income)The results may not be realistic due to assumptions underwhich the models are studied Indeed the validity of theassumption stipulating that price is constant seems to belimited to a small-scale fishery wildlife or forestry Alsothe assumption of free access to valuable resources must bediscarded because it inevitably leads to an overexploitationFor instance the Moroccan octopus crisis arising between1996 and 1997 generated a biological (overexploitation of

the resource) and economic instability (raising the resourceprice)

To avoid these situations Jerry and Raıssi [6] havestudied the biological and economic stabilizability by addinga dynamic prevailing on the market in order to take intoaccount resource price variations based on a standard open-access fisheries model (see [1 7ndash10]) the model was given by

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905

119889119901 (119905) = [119904 (119886 minus 119901 (119905) minus 119864119909 (119905))] 119889119905

(1)

where 119909(119905) is the population densities at time 119905119901(119905) is the unitprice of the population at time 119905 The constants 119903 gt 0 119896 gt 0

are the intrinsic growth rate and the carrying capacity of thepopulation respectively 119864 gt 0 is constant representing har-vesting effort 119886 is a positive constant parameter representingthe market capacity 119904 is the price speed adjustment and itreflects market competition (ie perfect elasticity of price)

2 Abstract and Applied Analysis

In daily life economic profit is a very important factorfor governments merchants and even every citizen so it isnecessary to investigate biological economic systems whichare always described by differential-algebraic equations Atpresent most of differential-algebraic equations can be foundin the general power systems [11 12] economic administra-tion [13] robot system [14] mechanical engineering [15] andso on However to the best of our knowledge the reports ofsuch systems are few in biology Considering the economictheory of fishery resource [16] proposed by Gordon in 1954this paper studies a class of singular biological economicmodel as follows

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905

119889119901 (119905) = [119904 (119886 minus 119901 (119905) minus 119864119909 (119905))] 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(2)

where 119888 gt 0 is harvesting cost per unit harvesting effortfor the population 119898(119905) ge 0 is the economic profit perunit harvesting effort at time 119905 From the previous modelingview the economic profit is variable In order to facilitatethe analysis many authors assume that the economic profitis constant (see [17ndash20]) In fact the economic profit is notalways constant and may vary with numerous factors suchas seasonality revenue market demand and harvesting costand so on Hence it is more reasonable that the economicprofit is a variable from real world point of view andmodel (2) is more suitable for the analysis of the dynamicbehavior of biological economic systemsThe economic profitis affected by the unit price of the population and thepopulation densities of our discussedmodel and other factorsare omitted However owing to difficulty arising in analysisfew results are concerned with such systems

It is well known that the white noise always exists andwe cannot omit the influence of the white noise to manydynamical systems In reality due to continuous fluctuationsin the environment (eg variation in intensity of sunlighttemperature water level etc) parameters involved inmodelsare not absolute constants but they always fluctuate aroundsome average value As a result the population density neverattains a fixed value with the advancement of time but ratherexhibits continuous oscillation around some average valuesBased upon these factors stochastic population models havereceived more and more attention [21ndash23] Many authorsstudied the effect of the stochastic perturbation to the biologi-cal economic systemwith different functional responses suchas [24ndash27] However only a small amount of work has beenstudied for the stochastic perturbation biological economicsystems with differential-algebraic equations In this paperwe introduce stochastic perturbation to system (2) and obtainthe following stochastic singular system

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905 + 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905))

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(3)

where 119882(119905) is one-dimensional Brownian motion with119882(0) = 0 and 1205902 is intensities of the white noise

The proposed system (3) has a more widely practicalbackground in this paper For example we show that thereis a stationary distribution of system (3) if the white noise issmall While if the white noise is large we prove that the preypopulation will either extinct or its distribution convergesto a probability measure So we may study the economicprofit through the changes of population density and thevariation of market price in the stochastic environment Forthis reason we say the stochastic model is more realistic thanthe deterministic model

The aim of this paper is to discuss the finite-time stabilityand control problem of the system (2) and (3) It is nowworthpointing out that the stability stabilization and control per-formancementioned previously concern the desired behaviorof the controlled dynamic systems over an infinite-timeinterval which always deals with the asymptotic property ofsystem trajectories [28ndash30] But in some practical processesthe asymptotically stable system over an infinite-time intervaldoes not mean that it has good transient characteristics forinstance biochemistry reaction system robot control systemcommunication network system and so forthTherefore it isnecessary to study the transient behavior over a finite-timeinterval

On the other hand the literature on finite-time stability(FTS) (or short-time stability) of systems has attracted par-ticular interests of researchers Some earlier results on finite-time stability date back to the 1960s [31] Recently the conceptof finite-time stability has been revisited in the light oflinear matrix inequality theory Several sufficient conditionsfor finite-time stability and stabilization of continuous timesystems or discrete time systems have been presented [3233] References [34 35] have extended the definition offinite-time stability to linear systems with impulsive effectsor singular systems with impulsive effects respectively andderived some sufficient conditions for finite-time stabilityand stabilization problems Very recently [36] generalizedthe results of [32] to linear stochastic systems Howeverto date and to the best of our knowledge no work so farhas been done on the problem of finite-time stability for aclass of stochastic singular biological economic systems Theproblems are important in many practice applications whichmotivate the main purpose of our study

T-S fuzzy model proposed by Takagi and Sugeno [37]in 1985 has been shown to be a powerful tool for modelingcomplex nonlinear systems It is wellknown that by means ofthe T-S fuzzy model a nonlinear system can be representedby a weighted sum of some simple linear subsystems and thencan be stabilized by a model-based fuzzy control Thereforemany issues related to the stability analysis of the complexnonlinear systems can be discussed using the T-S fuzzymodel

In this paper we investigate the problem of finite-time stability and control for stochastic singular biologicaleconomic systems Our results are totally different from thoseprevious results Firstly the system (2) and (3) are establishedfor fuzzy models based on T-S fuzzy control approach Thensufficient conditions of finite-time stability are obtained for

Abstract and Applied Analysis 3

the system (2) and (3) One of the contributions of this paperis that the economic profit which is a variable rather than aconstant of the biological economic systems is discussedWestudy the economic profit fluctuations through the unit pricerange and the population changes The other contribution isto study the finite-time stability for the biological economicsystem with stochastic perturbation and sufficient criterionsare presented for the solvability of the problem which canbe reduced to a feasibility problem involving restricted linearmatrix inequalities with a fixed parameter Based on this thestate feedback controller is designed so that the biologicalpopulation can be controlled in the bounded range by themanagement of the open resource The corresponding suf-ficient conditions are given as well Finally some numericalsimulations on feasible region are given to illustrate theeffectiveness of our approach

The rest of the paper is organized as follows In Section 2fuzzy modeling and preliminaries are stated in this paperIn Section 3 the finite-time stability of stochastic biologicaleconomic systems is studied which provides some sufficientconditions for systems to be FTS Meanwhile theorems ofsufficient conditions for FTS are established In Section 4finite-time controllers are designed for the stochastic biolog-ical economic systems Numerical examples are provided inSection 5 Conclusions are given in Section 6

Notations In this paper the superscript ldquoTrdquo stands for matrixtransposition The notation 119883 gt 0 means 119883 is a positivedefinite matrix 120576(119909) denotes the expectation of stochasticvariable 119909 deg(sdot) denotes the degree of the determinant Inaddition the symbol ldquolowastrdquo denotes the transposed elementsin the symmetric positions of a matrix and diag(sdot) standsfor a block-diagonal matrix 120582min(119875) and 120582max(119875) denote thesmallest and the largest eigenvalue of matrix 119875 respectively

2 Fuzzy Modeling and Preliminaries

Considering the model (2) for simplicity we use the follow-ing coordinate transformation

120585 (119905) = 119901 (119905) minus 119886 (4)

Then the model (2) can be rewritten as follows

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905

119889120585 (119905) = [minus119904119864119909 (119905) minus 119904120585 (119905)] 119889119905

0 = (119886 minus 119888) 119864119909 (119905) + 119864119909 (119905) 120585 (119905) minus 119898 (119905)

(5)

Let 119883(119905) = [119909(119905) 120585(119905) 119898(119905)]Τ the previous model can be

expressed by fuzzymodel which consists of the following tworules

1198771 If 119909 (119905) is 119865

1 then Ξ (119905) = 119860

1119883 (119905)

1198772 If 119909 (119905) is 119865

2 then Ξ (119905) = 119860

2119883 (119905)

(6)

where Ξ = [ 1 0 00 1 00 0 0

] 1198601= [2119903minus119864 0 0

minus119904119864 minus119904 0

119886119864minus119888119864 minus119896119864 minus1

] 1198602= [minus119864 0 0

minus119904119864 minus119904 0

119886119864minus119888119864 119896119864 minus1

]

119865119894(119894 = 1 2) is the fuzzy set ℎ

119894(119909(119905)) ge 0 is a membership

function of 119865119894 and ℎ

1(119909(119905)) = (12)(1 minus (119909(119905)119896)) ℎ

2(119909(119905)) =

(12)(1 + (119909(119905)119896)) and sum2119894=1ℎ119894(119909(119905)) = 1

By using a standard fuzzy singleton inference methoda singleton fuzzifier to produce a fuzzy inference and aweighted center-average defuzzifier we can obtain the globalfuzzy system as follows

Ξ (119905) = 119860ℎ119883 (119905) (7)

where 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894

As for the model (3) using the similar method we canobtain

Ξ119889119883 (119905) = 119860ℎ119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905) (8)

where 119862 = [120590 0 0

0 0 0

0 0 0

]

Definition 1 (i) The pair (Ξ 119860ℎ) is said to be regular if

det(120582Ξ minus 119860ℎ) is not identically zero

(ii) The pair (Ξ 119860ℎ) is said to be impulse free if

deg(det(120582Ξ minus 119860ℎ)) = rank(Ξ)

Remark 2 In this paper through the calculation it is knownthat the pair (Ξ 119860

ℎ) is regular

Definition 3 Given two positive real numbers 1198881 1198882with 119888

1le

1198882 a positive definite matrix 119877 system (7) is said to be finite-

time stable with respect to (1198881 1198882 119879 119877) if

119883Τ(0) ΞΤ119877Ξ119883 (0) le 119888

1997904rArr 119883

Τ(119905) ΞΤ119877Ξ119883 (119905) lt 119888

2

forall119905 isin [0 119879]

(9)

Definition 4 Given two positive real numbers 1198881 1198882with

1198881le 1198882 a positive definite matrix 119877 system (8) is said to be

stochastic finite-time stable with respect to (1198881 1198882 119879 119877) if

120576 119883Τ(0) ΞΤ119877Ξ119883 (0) le 119888

1997904rArr 120576 119883

Τ(119905) ΞΤ119877Ξ119883 (119905) lt 119888

2

forall119905 isin [0 119879]

(10)

Remark 5 Asymptotic stability and FTS are independentconcepts of each other as it was described in [31] Incertain cases asymptotic stability can be also consideredas an additional requirement while restricting our attentionover a finite-time interval The definition of FTS can beinterpreted in terms of ellipsoidal domains The set definedby 119883Τ(0)ΞΤ119877Ξ119883(0) le 119888

1 contains all the admissible initial

states Instead the inequality 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 1198882 defines

a time-varying ellipsoid which bounds the state trajectoryover the finite-time interval 119905 isin [0 119879] In addition wecan fix the upper bound beforehand in finite-time controland in general the initial and ultimates bound are not thesame Thus this concept will be more meaningful in prac-tice such as aircraft control networked control and robotoperating

4 Abstract and Applied Analysis

Definition 6 (see [38]) Let 119881(119909(119905) 119905) be the stochastic Lya-punov function of the stochastic system where 119909(119905) satisfiesthe following stochastic differential equation

119889119909 (119905) = 119891 (119905) 119889119905 + 119892 (119905) 119889119882 (119905) (11)

Define a weak infinitesimal operator L of random process119909(119905) 119905 gt 0

L119881 (119909 (119905) 119905) = 119881119905(119909 (119905) 119905) + 119881

119909(119909 (119905) 119905) 119891 (119905)

+1

2tr [119892Τ (119905) 119881

119909119909(119909 (119905) 119905) 119892 (119905)]

(12)

Lemma 7 (see [39] Gronwalls inequality) Let V(119905) be anonnegative function such that

V (119905) le 119886 + 119887int119905

0

V (119904) 119889119904 0 le 119905 le 119879 (13)

for some constants 119886 119887 ge 0 then we have

V (119905) le 119886 exp (119887119905) 0 le 119905 le 119879 (14)

Lemma 8 (Schurrsquos complement) For real matrices 119873119872119879 =119872 and 119877 = 119877

119879gt 0 the following three conditions are

equivalent

(1) 119872+119873119877minus1119873119879lt 0

(2) ( 119872 119873119873119879minus119877) lt 0

(3) ( 119872 minus119873minus119873119879minus119877) lt 0

Lemma 9 (see [40]) The following items are true

(i) Assume that rank(Ξ) = 119903 there exist two orthogonalmatrices119880 and119881 such that Ξ has the decomposition as

Ξ = 119880[Σ1199030

lowast 0]119881119879= 119880[

1198681199030

lowast 0] V119879 (15)

where Σ119903= diag120575

1 1205752 120575

119903 with 120575

119896gt 0 for all 119896 =

1 2 119903 Partition 119880 = [1198801 1198802] 119881 = [1198811 1198812] andV = [1198811Σ119903 1198812] with Ξ1198812 = 0 and 119880Τ2 Ξ = 0

(ii) If P satisfies

Ξ119875Τ= 119875ΞΤge 0 (16)

Then = 119880Τ119875V with119880 and V satisfying (8) if and onlyif

= [11987511

11987512

0 11987522

] (17)

with 11987511

ge 0 isin R119903times119903 In addition when P isnonsingular we have 119875

11gt 0 and det(119875

22) = 0

Furthermore 119875 satisfying (16) can be parameterized as

119875 = ΞVminusΤ119884Vminus1 + 119880119885119881Τ2 (18)

where 119884 = diag11987511 Ψ 119885 = [119875

Τ

12119875Τ

22]Τ

and Ψ isin

R(119899minus119903)times(119899minus119903) is an arbitrary parameter matrix

(iii) If P is a nonsingular matrix R and Ψ are two positivedefinite matrices P and Ξ satisfy (16) Y is a diagonalmatrix from (18) and the following equality holds

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (19)

Then the positive definite matrix 119876 = 119877minus12

119880119884minus1119880Τ

119877minus12 is a solution of (19)

3 FTS of the Stochastic Singular BiologicalEconomic System

This section provides finite-time stability analysis results forthe stochastic singular biological economic system somenovel sufficient conditions for finite-time stability of thestochastic singular system are given Through the previousanalysis we know that the system (2) and (3) are equivalentto the system (7) and (8) respectively Because the system (7)and (8) are regular the following theorems are given

31 FTS of the System (7)

Theorem 10 For a given time constant119879 gt 0 and a scalar 120572 gt0 the system (7) is finite-time stable with respect to (119888

1 1198882 119879 119877)

if there exist a nonsingular matrix 119875 and a symmetric positivedefinite matrix 119876 such that the following conditions hold

119875ΞΤ= Ξ119875Τge 0 (20)

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τlt 0 (21a)

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τlt 0 (21b)

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (22)

120582max (119876) 1198881119890120572119879minus 1198882120582min (119876) lt 0 (23)

Proof Consider the following Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (24)

We first prove that the following condition (25) is equivalentto (21a) and (21b)

(119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (25)

When 119905 isin [0 119879] we calculate the time derivative of119881(119883(119905) 119905)along the solution of system (7) and obtain

(119883 (119905) 119905) = 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎ)119883 (119905) (26)

which leads to

(119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ)119883 (119905)

(27)

Abstract and Applied Analysis 5

So (27) is equivalent to the following inequality

119860ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ lt 0 (28)

Pre- and postmultiplying (28) by 119875 and 119875Τ we get

119860ℎ119875Τ+ 119875119860Τ

ℎminus 120572Ξ119875

Τlt 0 (29)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decomposi-

tion we can see that (29) is equivalent to (21a) and (21b)Thus dividing by 119881(119883(119905) 119905) on both sides of (25) and

integrating it from 0 to 119905 with 119905 isin [0 119879] one gets

119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (30)

By condition (22) one can deduce that

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905)

= 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876)119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119909 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(31)

From (31) it is easy to obtain

119883Τ(119905) ΞΤ119877Ξ119883 (119905) le

120582max (119876)

120582min (119876)1198881119890120572119879 (32)

Considering condition (23) and inequality (32) it is easy toget 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes the

proof of the theorem

32 FTS of the System (8)

Theorem 11 For a given time constant 119879 gt 0 and a scalar120572 gt 0 the system (8) is stochastic finite-time stable with respectto (1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 and a

symmetric positive definite matrix 119876 such that (20) (22) and(23) hold and the following matrix inequalities are satisfied

[

[

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33a)

[

[

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33b)

where 119876 = 119877minus12

119876minus1119877minus12

Proof Let us consider the quadratic Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (34)

LetL be the infinitesimal generator (also called the averagedderivative) We first show the following condition

L119881 (119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (35)

Now we will prove that condition (35) is equivalent to (33a)and (33b) Applying Itorsquos formula it follows that

L119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862)119883 (119905)

(36)

which leads to

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ

+119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(37)

Combining condition (22) with 119876 = 119877minus12

119876minus1119877minus12 we can

obtain

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ

+119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(38)

So (37) is equivalent to

119860Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ lt 0 (39)

Pre- and postmultiplying (39) by 119875 and 119875Τ we get

119875119860Τ

ℎ+ 119860ℎ119875Τ+ 119875119862ΤΞΤ119876minus1Ξ119862119875Τminus 120572Ξ119875

Τlt 0 (40)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decompo-

sition and by Lemma 8 we can see that (40) is equivalent to(33a) and (33b)

Hence integrating both sides of (35) from 0 to 119905 with 119905 isin[0 119879] and then taking the expectation it yields

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) + 120572int

119905

0

120576 119881 (119883 (119904) 119904) 119889119904

(41)

By Lemma 7 we can obtain

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (42)

From (34) it follows that

120576 119881 (119883 (119905) 119905) = 120576 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876) 120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119883 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(43)

6 Abstract and Applied Analysis

From (43) it is easy to obtain

120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905) lt

120582max (119876)

120582min (119876)1198881119890120572119879 (44)

Considering condition (23) and inequality (44) it is easy toget 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes

the proof of the theorem

4 Finite-Time Control of the StochasticSingular Biological Economic System

Unstable fluctuations have always been regarded as unfa-vorable ones from the ecological managersrsquo point of viewMeanwhile price variation depends on the gap betweenthe demand function and supply function In order to planharvesting strategies and maintain the sustainable develop-ment of system it is necessary to take action to stabilizebiological population and market directed economy Thispaper proposes the following state feedback control methodfor the system (2) and (3)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(45)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

+ 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(46)

where 119906(119905) is control variable representing the managementof the open resource

Then we use the similar T-S fuzzy method shown inSection 2 and consider the following state feedback fuzzycontroller

119906 (119905) =

2

sum

119894=1

ℎ119894(119909 (119905)) 119871

119894119883 (119905) (47)

where 119871119894is the state feedback gain to be designed

Themodel (45) and (46) can be expressed by the followingfuzzy models

Ξ (119905) =

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905)) (119860

119894+ 119861119871119895)119883 (119905) (48)

Ξ119889119883 (119905)

=

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905))

times [(119860119894+ 119861119871119895)119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905)]

(49)

where 119861 = [1 0 0]Τ

In the sequence we provide finite-time stabilizationresults for above the mentioned closed-loop system (48) and(49) At the same time some novel sufficient conditions forfinite-time stabilization of the closed-loop singular biologicaleconomic system are given

41 Finite-Time Control of the System (45)

Theorem 12 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (48) is finite-time stable with respect to(1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 a symmetric

positive definite matrix 119876 and matrix 119884119895 119895 = 1 2 such

that (22) (23) hold and the following matrix inequalities aresatisfied

119875ΞΤ= Ξ119875Τge 0 (50)

Φ119894119894lt 0 119894 = 119895 119894 119895 = 1 2 (51a)

Φ119894119895+ Φ119895119894lt 0 119894 lt 119895 119894 119895 = 1 2 (51b)

where Φ119894119895= 119875119860Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ Then thedesired state feedback gain can be chosen as 119871

119895= 119884119895119875minusΤ

Proof Considering Theorem 10 and the closed-loop system(48) we can obtain the following matrix inequality

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) Π

119894119895lt 0 (52)

where Π119894119895= (119860119894+ 119861119871119895)119875Τ+ 119875(119860

119894+ 119861119871119895)Τminus 120572Ξ119875

ΤLet 119884119895= 119871119895119875Τ Then

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905))

times (119875119860Τ

119894+ 119884Τ

119895119861119879+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ) lt 0

(53)

Obviously (53) is equivalent to

2

sum

119894=1

ℎ2

119894(119909 (119905)) Φ

119894119894+

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) (Φ

119894119895+ Φ119895119894) lt 0

(54)

The above inequality (54) is equivalent to (51a) and (51b)Then by a similar argument in the proof of Theorem 10

and using (22)-(23) we can easily get the above results Thiscompletes the proof of the theorem

42 Finite-Time Control of the System (46)

Theorem 13 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (49) is stochastic finite-time stable withrespect to (119888

1 1198882 119879 119877) if there exist a nonsingular matrix 119875 a

symmetric positive definite matrix 119876 and matrix 119884119895 119895 = 1 2

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

In daily life economic profit is a very important factorfor governments merchants and even every citizen so it isnecessary to investigate biological economic systems whichare always described by differential-algebraic equations Atpresent most of differential-algebraic equations can be foundin the general power systems [11 12] economic administra-tion [13] robot system [14] mechanical engineering [15] andso on However to the best of our knowledge the reports ofsuch systems are few in biology Considering the economictheory of fishery resource [16] proposed by Gordon in 1954this paper studies a class of singular biological economicmodel as follows

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905

119889119901 (119905) = [119904 (119886 minus 119901 (119905) minus 119864119909 (119905))] 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(2)

where 119888 gt 0 is harvesting cost per unit harvesting effortfor the population 119898(119905) ge 0 is the economic profit perunit harvesting effort at time 119905 From the previous modelingview the economic profit is variable In order to facilitatethe analysis many authors assume that the economic profitis constant (see [17ndash20]) In fact the economic profit is notalways constant and may vary with numerous factors suchas seasonality revenue market demand and harvesting costand so on Hence it is more reasonable that the economicprofit is a variable from real world point of view andmodel (2) is more suitable for the analysis of the dynamicbehavior of biological economic systemsThe economic profitis affected by the unit price of the population and thepopulation densities of our discussedmodel and other factorsare omitted However owing to difficulty arising in analysisfew results are concerned with such systems

It is well known that the white noise always exists andwe cannot omit the influence of the white noise to manydynamical systems In reality due to continuous fluctuationsin the environment (eg variation in intensity of sunlighttemperature water level etc) parameters involved inmodelsare not absolute constants but they always fluctuate aroundsome average value As a result the population density neverattains a fixed value with the advancement of time but ratherexhibits continuous oscillation around some average valuesBased upon these factors stochastic population models havereceived more and more attention [21ndash23] Many authorsstudied the effect of the stochastic perturbation to the biologi-cal economic systemwith different functional responses suchas [24ndash27] However only a small amount of work has beenstudied for the stochastic perturbation biological economicsystems with differential-algebraic equations In this paperwe introduce stochastic perturbation to system (2) and obtainthe following stochastic singular system

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905 + 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905))

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(3)

where 119882(119905) is one-dimensional Brownian motion with119882(0) = 0 and 1205902 is intensities of the white noise

The proposed system (3) has a more widely practicalbackground in this paper For example we show that thereis a stationary distribution of system (3) if the white noise issmall While if the white noise is large we prove that the preypopulation will either extinct or its distribution convergesto a probability measure So we may study the economicprofit through the changes of population density and thevariation of market price in the stochastic environment Forthis reason we say the stochastic model is more realistic thanthe deterministic model

The aim of this paper is to discuss the finite-time stabilityand control problem of the system (2) and (3) It is nowworthpointing out that the stability stabilization and control per-formancementioned previously concern the desired behaviorof the controlled dynamic systems over an infinite-timeinterval which always deals with the asymptotic property ofsystem trajectories [28ndash30] But in some practical processesthe asymptotically stable system over an infinite-time intervaldoes not mean that it has good transient characteristics forinstance biochemistry reaction system robot control systemcommunication network system and so forthTherefore it isnecessary to study the transient behavior over a finite-timeinterval

On the other hand the literature on finite-time stability(FTS) (or short-time stability) of systems has attracted par-ticular interests of researchers Some earlier results on finite-time stability date back to the 1960s [31] Recently the conceptof finite-time stability has been revisited in the light oflinear matrix inequality theory Several sufficient conditionsfor finite-time stability and stabilization of continuous timesystems or discrete time systems have been presented [3233] References [34 35] have extended the definition offinite-time stability to linear systems with impulsive effectsor singular systems with impulsive effects respectively andderived some sufficient conditions for finite-time stabilityand stabilization problems Very recently [36] generalizedthe results of [32] to linear stochastic systems Howeverto date and to the best of our knowledge no work so farhas been done on the problem of finite-time stability for aclass of stochastic singular biological economic systems Theproblems are important in many practice applications whichmotivate the main purpose of our study

T-S fuzzy model proposed by Takagi and Sugeno [37]in 1985 has been shown to be a powerful tool for modelingcomplex nonlinear systems It is wellknown that by means ofthe T-S fuzzy model a nonlinear system can be representedby a weighted sum of some simple linear subsystems and thencan be stabilized by a model-based fuzzy control Thereforemany issues related to the stability analysis of the complexnonlinear systems can be discussed using the T-S fuzzymodel

In this paper we investigate the problem of finite-time stability and control for stochastic singular biologicaleconomic systems Our results are totally different from thoseprevious results Firstly the system (2) and (3) are establishedfor fuzzy models based on T-S fuzzy control approach Thensufficient conditions of finite-time stability are obtained for

Abstract and Applied Analysis 3

the system (2) and (3) One of the contributions of this paperis that the economic profit which is a variable rather than aconstant of the biological economic systems is discussedWestudy the economic profit fluctuations through the unit pricerange and the population changes The other contribution isto study the finite-time stability for the biological economicsystem with stochastic perturbation and sufficient criterionsare presented for the solvability of the problem which canbe reduced to a feasibility problem involving restricted linearmatrix inequalities with a fixed parameter Based on this thestate feedback controller is designed so that the biologicalpopulation can be controlled in the bounded range by themanagement of the open resource The corresponding suf-ficient conditions are given as well Finally some numericalsimulations on feasible region are given to illustrate theeffectiveness of our approach

The rest of the paper is organized as follows In Section 2fuzzy modeling and preliminaries are stated in this paperIn Section 3 the finite-time stability of stochastic biologicaleconomic systems is studied which provides some sufficientconditions for systems to be FTS Meanwhile theorems ofsufficient conditions for FTS are established In Section 4finite-time controllers are designed for the stochastic biolog-ical economic systems Numerical examples are provided inSection 5 Conclusions are given in Section 6

Notations In this paper the superscript ldquoTrdquo stands for matrixtransposition The notation 119883 gt 0 means 119883 is a positivedefinite matrix 120576(119909) denotes the expectation of stochasticvariable 119909 deg(sdot) denotes the degree of the determinant Inaddition the symbol ldquolowastrdquo denotes the transposed elementsin the symmetric positions of a matrix and diag(sdot) standsfor a block-diagonal matrix 120582min(119875) and 120582max(119875) denote thesmallest and the largest eigenvalue of matrix 119875 respectively

2 Fuzzy Modeling and Preliminaries

Considering the model (2) for simplicity we use the follow-ing coordinate transformation

120585 (119905) = 119901 (119905) minus 119886 (4)

Then the model (2) can be rewritten as follows

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905

119889120585 (119905) = [minus119904119864119909 (119905) minus 119904120585 (119905)] 119889119905

0 = (119886 minus 119888) 119864119909 (119905) + 119864119909 (119905) 120585 (119905) minus 119898 (119905)

(5)

Let 119883(119905) = [119909(119905) 120585(119905) 119898(119905)]Τ the previous model can be

expressed by fuzzymodel which consists of the following tworules

1198771 If 119909 (119905) is 119865

1 then Ξ (119905) = 119860

1119883 (119905)

1198772 If 119909 (119905) is 119865

2 then Ξ (119905) = 119860

2119883 (119905)

(6)

where Ξ = [ 1 0 00 1 00 0 0

] 1198601= [2119903minus119864 0 0

minus119904119864 minus119904 0

119886119864minus119888119864 minus119896119864 minus1

] 1198602= [minus119864 0 0

minus119904119864 minus119904 0

119886119864minus119888119864 119896119864 minus1

]

119865119894(119894 = 1 2) is the fuzzy set ℎ

119894(119909(119905)) ge 0 is a membership

function of 119865119894 and ℎ

1(119909(119905)) = (12)(1 minus (119909(119905)119896)) ℎ

2(119909(119905)) =

(12)(1 + (119909(119905)119896)) and sum2119894=1ℎ119894(119909(119905)) = 1

By using a standard fuzzy singleton inference methoda singleton fuzzifier to produce a fuzzy inference and aweighted center-average defuzzifier we can obtain the globalfuzzy system as follows

Ξ (119905) = 119860ℎ119883 (119905) (7)

where 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894

As for the model (3) using the similar method we canobtain

Ξ119889119883 (119905) = 119860ℎ119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905) (8)

where 119862 = [120590 0 0

0 0 0

0 0 0

]

Definition 1 (i) The pair (Ξ 119860ℎ) is said to be regular if

det(120582Ξ minus 119860ℎ) is not identically zero

(ii) The pair (Ξ 119860ℎ) is said to be impulse free if

deg(det(120582Ξ minus 119860ℎ)) = rank(Ξ)

Remark 2 In this paper through the calculation it is knownthat the pair (Ξ 119860

ℎ) is regular

Definition 3 Given two positive real numbers 1198881 1198882with 119888

1le

1198882 a positive definite matrix 119877 system (7) is said to be finite-

time stable with respect to (1198881 1198882 119879 119877) if

119883Τ(0) ΞΤ119877Ξ119883 (0) le 119888

1997904rArr 119883

Τ(119905) ΞΤ119877Ξ119883 (119905) lt 119888

2

forall119905 isin [0 119879]

(9)

Definition 4 Given two positive real numbers 1198881 1198882with

1198881le 1198882 a positive definite matrix 119877 system (8) is said to be

stochastic finite-time stable with respect to (1198881 1198882 119879 119877) if

120576 119883Τ(0) ΞΤ119877Ξ119883 (0) le 119888

1997904rArr 120576 119883

Τ(119905) ΞΤ119877Ξ119883 (119905) lt 119888

2

forall119905 isin [0 119879]

(10)

Remark 5 Asymptotic stability and FTS are independentconcepts of each other as it was described in [31] Incertain cases asymptotic stability can be also consideredas an additional requirement while restricting our attentionover a finite-time interval The definition of FTS can beinterpreted in terms of ellipsoidal domains The set definedby 119883Τ(0)ΞΤ119877Ξ119883(0) le 119888

1 contains all the admissible initial

states Instead the inequality 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 1198882 defines

a time-varying ellipsoid which bounds the state trajectoryover the finite-time interval 119905 isin [0 119879] In addition wecan fix the upper bound beforehand in finite-time controland in general the initial and ultimates bound are not thesame Thus this concept will be more meaningful in prac-tice such as aircraft control networked control and robotoperating

4 Abstract and Applied Analysis

Definition 6 (see [38]) Let 119881(119909(119905) 119905) be the stochastic Lya-punov function of the stochastic system where 119909(119905) satisfiesthe following stochastic differential equation

119889119909 (119905) = 119891 (119905) 119889119905 + 119892 (119905) 119889119882 (119905) (11)

Define a weak infinitesimal operator L of random process119909(119905) 119905 gt 0

L119881 (119909 (119905) 119905) = 119881119905(119909 (119905) 119905) + 119881

119909(119909 (119905) 119905) 119891 (119905)

+1

2tr [119892Τ (119905) 119881

119909119909(119909 (119905) 119905) 119892 (119905)]

(12)

Lemma 7 (see [39] Gronwalls inequality) Let V(119905) be anonnegative function such that

V (119905) le 119886 + 119887int119905

0

V (119904) 119889119904 0 le 119905 le 119879 (13)

for some constants 119886 119887 ge 0 then we have

V (119905) le 119886 exp (119887119905) 0 le 119905 le 119879 (14)

Lemma 8 (Schurrsquos complement) For real matrices 119873119872119879 =119872 and 119877 = 119877

119879gt 0 the following three conditions are

equivalent

(1) 119872+119873119877minus1119873119879lt 0

(2) ( 119872 119873119873119879minus119877) lt 0

(3) ( 119872 minus119873minus119873119879minus119877) lt 0

Lemma 9 (see [40]) The following items are true

(i) Assume that rank(Ξ) = 119903 there exist two orthogonalmatrices119880 and119881 such that Ξ has the decomposition as

Ξ = 119880[Σ1199030

lowast 0]119881119879= 119880[

1198681199030

lowast 0] V119879 (15)

where Σ119903= diag120575

1 1205752 120575

119903 with 120575

119896gt 0 for all 119896 =

1 2 119903 Partition 119880 = [1198801 1198802] 119881 = [1198811 1198812] andV = [1198811Σ119903 1198812] with Ξ1198812 = 0 and 119880Τ2 Ξ = 0

(ii) If P satisfies

Ξ119875Τ= 119875ΞΤge 0 (16)

Then = 119880Τ119875V with119880 and V satisfying (8) if and onlyif

= [11987511

11987512

0 11987522

] (17)

with 11987511

ge 0 isin R119903times119903 In addition when P isnonsingular we have 119875

11gt 0 and det(119875

22) = 0

Furthermore 119875 satisfying (16) can be parameterized as

119875 = ΞVminusΤ119884Vminus1 + 119880119885119881Τ2 (18)

where 119884 = diag11987511 Ψ 119885 = [119875

Τ

12119875Τ

22]Τ

and Ψ isin

R(119899minus119903)times(119899minus119903) is an arbitrary parameter matrix

(iii) If P is a nonsingular matrix R and Ψ are two positivedefinite matrices P and Ξ satisfy (16) Y is a diagonalmatrix from (18) and the following equality holds

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (19)

Then the positive definite matrix 119876 = 119877minus12

119880119884minus1119880Τ

119877minus12 is a solution of (19)

3 FTS of the Stochastic Singular BiologicalEconomic System

This section provides finite-time stability analysis results forthe stochastic singular biological economic system somenovel sufficient conditions for finite-time stability of thestochastic singular system are given Through the previousanalysis we know that the system (2) and (3) are equivalentto the system (7) and (8) respectively Because the system (7)and (8) are regular the following theorems are given

31 FTS of the System (7)

Theorem 10 For a given time constant119879 gt 0 and a scalar 120572 gt0 the system (7) is finite-time stable with respect to (119888

1 1198882 119879 119877)

if there exist a nonsingular matrix 119875 and a symmetric positivedefinite matrix 119876 such that the following conditions hold

119875ΞΤ= Ξ119875Τge 0 (20)

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τlt 0 (21a)

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τlt 0 (21b)

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (22)

120582max (119876) 1198881119890120572119879minus 1198882120582min (119876) lt 0 (23)

Proof Consider the following Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (24)

We first prove that the following condition (25) is equivalentto (21a) and (21b)

(119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (25)

When 119905 isin [0 119879] we calculate the time derivative of119881(119883(119905) 119905)along the solution of system (7) and obtain

(119883 (119905) 119905) = 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎ)119883 (119905) (26)

which leads to

(119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ)119883 (119905)

(27)

Abstract and Applied Analysis 5

So (27) is equivalent to the following inequality

119860ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ lt 0 (28)

Pre- and postmultiplying (28) by 119875 and 119875Τ we get

119860ℎ119875Τ+ 119875119860Τ

ℎminus 120572Ξ119875

Τlt 0 (29)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decomposi-

tion we can see that (29) is equivalent to (21a) and (21b)Thus dividing by 119881(119883(119905) 119905) on both sides of (25) and

integrating it from 0 to 119905 with 119905 isin [0 119879] one gets

119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (30)

By condition (22) one can deduce that

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905)

= 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876)119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119909 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(31)

From (31) it is easy to obtain

119883Τ(119905) ΞΤ119877Ξ119883 (119905) le

120582max (119876)

120582min (119876)1198881119890120572119879 (32)

Considering condition (23) and inequality (32) it is easy toget 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes the

proof of the theorem

32 FTS of the System (8)

Theorem 11 For a given time constant 119879 gt 0 and a scalar120572 gt 0 the system (8) is stochastic finite-time stable with respectto (1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 and a

symmetric positive definite matrix 119876 such that (20) (22) and(23) hold and the following matrix inequalities are satisfied

[

[

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33a)

[

[

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33b)

where 119876 = 119877minus12

119876minus1119877minus12

Proof Let us consider the quadratic Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (34)

LetL be the infinitesimal generator (also called the averagedderivative) We first show the following condition

L119881 (119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (35)

Now we will prove that condition (35) is equivalent to (33a)and (33b) Applying Itorsquos formula it follows that

L119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862)119883 (119905)

(36)

which leads to

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ

+119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(37)

Combining condition (22) with 119876 = 119877minus12

119876minus1119877minus12 we can

obtain

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ

+119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(38)

So (37) is equivalent to

119860Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ lt 0 (39)

Pre- and postmultiplying (39) by 119875 and 119875Τ we get

119875119860Τ

ℎ+ 119860ℎ119875Τ+ 119875119862ΤΞΤ119876minus1Ξ119862119875Τminus 120572Ξ119875

Τlt 0 (40)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decompo-

sition and by Lemma 8 we can see that (40) is equivalent to(33a) and (33b)

Hence integrating both sides of (35) from 0 to 119905 with 119905 isin[0 119879] and then taking the expectation it yields

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) + 120572int

119905

0

120576 119881 (119883 (119904) 119904) 119889119904

(41)

By Lemma 7 we can obtain

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (42)

From (34) it follows that

120576 119881 (119883 (119905) 119905) = 120576 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876) 120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119883 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(43)

6 Abstract and Applied Analysis

From (43) it is easy to obtain

120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905) lt

120582max (119876)

120582min (119876)1198881119890120572119879 (44)

Considering condition (23) and inequality (44) it is easy toget 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes

the proof of the theorem

4 Finite-Time Control of the StochasticSingular Biological Economic System

Unstable fluctuations have always been regarded as unfa-vorable ones from the ecological managersrsquo point of viewMeanwhile price variation depends on the gap betweenthe demand function and supply function In order to planharvesting strategies and maintain the sustainable develop-ment of system it is necessary to take action to stabilizebiological population and market directed economy Thispaper proposes the following state feedback control methodfor the system (2) and (3)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(45)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

+ 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(46)

where 119906(119905) is control variable representing the managementof the open resource

Then we use the similar T-S fuzzy method shown inSection 2 and consider the following state feedback fuzzycontroller

119906 (119905) =

2

sum

119894=1

ℎ119894(119909 (119905)) 119871

119894119883 (119905) (47)

where 119871119894is the state feedback gain to be designed

Themodel (45) and (46) can be expressed by the followingfuzzy models

Ξ (119905) =

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905)) (119860

119894+ 119861119871119895)119883 (119905) (48)

Ξ119889119883 (119905)

=

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905))

times [(119860119894+ 119861119871119895)119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905)]

(49)

where 119861 = [1 0 0]Τ

In the sequence we provide finite-time stabilizationresults for above the mentioned closed-loop system (48) and(49) At the same time some novel sufficient conditions forfinite-time stabilization of the closed-loop singular biologicaleconomic system are given

41 Finite-Time Control of the System (45)

Theorem 12 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (48) is finite-time stable with respect to(1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 a symmetric

positive definite matrix 119876 and matrix 119884119895 119895 = 1 2 such

that (22) (23) hold and the following matrix inequalities aresatisfied

119875ΞΤ= Ξ119875Τge 0 (50)

Φ119894119894lt 0 119894 = 119895 119894 119895 = 1 2 (51a)

Φ119894119895+ Φ119895119894lt 0 119894 lt 119895 119894 119895 = 1 2 (51b)

where Φ119894119895= 119875119860Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ Then thedesired state feedback gain can be chosen as 119871

119895= 119884119895119875minusΤ

Proof Considering Theorem 10 and the closed-loop system(48) we can obtain the following matrix inequality

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) Π

119894119895lt 0 (52)

where Π119894119895= (119860119894+ 119861119871119895)119875Τ+ 119875(119860

119894+ 119861119871119895)Τminus 120572Ξ119875

ΤLet 119884119895= 119871119895119875Τ Then

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905))

times (119875119860Τ

119894+ 119884Τ

119895119861119879+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ) lt 0

(53)

Obviously (53) is equivalent to

2

sum

119894=1

ℎ2

119894(119909 (119905)) Φ

119894119894+

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) (Φ

119894119895+ Φ119895119894) lt 0

(54)

The above inequality (54) is equivalent to (51a) and (51b)Then by a similar argument in the proof of Theorem 10

and using (22)-(23) we can easily get the above results Thiscompletes the proof of the theorem

42 Finite-Time Control of the System (46)

Theorem 13 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (49) is stochastic finite-time stable withrespect to (119888

1 1198882 119879 119877) if there exist a nonsingular matrix 119875 a

symmetric positive definite matrix 119876 and matrix 119884119895 119895 = 1 2

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

the system (2) and (3) One of the contributions of this paperis that the economic profit which is a variable rather than aconstant of the biological economic systems is discussedWestudy the economic profit fluctuations through the unit pricerange and the population changes The other contribution isto study the finite-time stability for the biological economicsystem with stochastic perturbation and sufficient criterionsare presented for the solvability of the problem which canbe reduced to a feasibility problem involving restricted linearmatrix inequalities with a fixed parameter Based on this thestate feedback controller is designed so that the biologicalpopulation can be controlled in the bounded range by themanagement of the open resource The corresponding suf-ficient conditions are given as well Finally some numericalsimulations on feasible region are given to illustrate theeffectiveness of our approach

The rest of the paper is organized as follows In Section 2fuzzy modeling and preliminaries are stated in this paperIn Section 3 the finite-time stability of stochastic biologicaleconomic systems is studied which provides some sufficientconditions for systems to be FTS Meanwhile theorems ofsufficient conditions for FTS are established In Section 4finite-time controllers are designed for the stochastic biolog-ical economic systems Numerical examples are provided inSection 5 Conclusions are given in Section 6

Notations In this paper the superscript ldquoTrdquo stands for matrixtransposition The notation 119883 gt 0 means 119883 is a positivedefinite matrix 120576(119909) denotes the expectation of stochasticvariable 119909 deg(sdot) denotes the degree of the determinant Inaddition the symbol ldquolowastrdquo denotes the transposed elementsin the symmetric positions of a matrix and diag(sdot) standsfor a block-diagonal matrix 120582min(119875) and 120582max(119875) denote thesmallest and the largest eigenvalue of matrix 119875 respectively

2 Fuzzy Modeling and Preliminaries

Considering the model (2) for simplicity we use the follow-ing coordinate transformation

120585 (119905) = 119901 (119905) minus 119886 (4)

Then the model (2) can be rewritten as follows

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905)] 119889119905

119889120585 (119905) = [minus119904119864119909 (119905) minus 119904120585 (119905)] 119889119905

0 = (119886 minus 119888) 119864119909 (119905) + 119864119909 (119905) 120585 (119905) minus 119898 (119905)

(5)

Let 119883(119905) = [119909(119905) 120585(119905) 119898(119905)]Τ the previous model can be

expressed by fuzzymodel which consists of the following tworules

1198771 If 119909 (119905) is 119865

1 then Ξ (119905) = 119860

1119883 (119905)

1198772 If 119909 (119905) is 119865

2 then Ξ (119905) = 119860

2119883 (119905)

(6)

where Ξ = [ 1 0 00 1 00 0 0

] 1198601= [2119903minus119864 0 0

minus119904119864 minus119904 0

119886119864minus119888119864 minus119896119864 minus1

] 1198602= [minus119864 0 0

minus119904119864 minus119904 0

119886119864minus119888119864 119896119864 minus1

]

119865119894(119894 = 1 2) is the fuzzy set ℎ

119894(119909(119905)) ge 0 is a membership

function of 119865119894 and ℎ

1(119909(119905)) = (12)(1 minus (119909(119905)119896)) ℎ

2(119909(119905)) =

(12)(1 + (119909(119905)119896)) and sum2119894=1ℎ119894(119909(119905)) = 1

By using a standard fuzzy singleton inference methoda singleton fuzzifier to produce a fuzzy inference and aweighted center-average defuzzifier we can obtain the globalfuzzy system as follows

Ξ (119905) = 119860ℎ119883 (119905) (7)

where 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894

As for the model (3) using the similar method we canobtain

Ξ119889119883 (119905) = 119860ℎ119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905) (8)

where 119862 = [120590 0 0

0 0 0

0 0 0

]

Definition 1 (i) The pair (Ξ 119860ℎ) is said to be regular if

det(120582Ξ minus 119860ℎ) is not identically zero

(ii) The pair (Ξ 119860ℎ) is said to be impulse free if

deg(det(120582Ξ minus 119860ℎ)) = rank(Ξ)

Remark 2 In this paper through the calculation it is knownthat the pair (Ξ 119860

ℎ) is regular

Definition 3 Given two positive real numbers 1198881 1198882with 119888

1le

1198882 a positive definite matrix 119877 system (7) is said to be finite-

time stable with respect to (1198881 1198882 119879 119877) if

119883Τ(0) ΞΤ119877Ξ119883 (0) le 119888

1997904rArr 119883

Τ(119905) ΞΤ119877Ξ119883 (119905) lt 119888

2

forall119905 isin [0 119879]

(9)

Definition 4 Given two positive real numbers 1198881 1198882with

1198881le 1198882 a positive definite matrix 119877 system (8) is said to be

stochastic finite-time stable with respect to (1198881 1198882 119879 119877) if

120576 119883Τ(0) ΞΤ119877Ξ119883 (0) le 119888

1997904rArr 120576 119883

Τ(119905) ΞΤ119877Ξ119883 (119905) lt 119888

2

forall119905 isin [0 119879]

(10)

Remark 5 Asymptotic stability and FTS are independentconcepts of each other as it was described in [31] Incertain cases asymptotic stability can be also consideredas an additional requirement while restricting our attentionover a finite-time interval The definition of FTS can beinterpreted in terms of ellipsoidal domains The set definedby 119883Τ(0)ΞΤ119877Ξ119883(0) le 119888

1 contains all the admissible initial

states Instead the inequality 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 1198882 defines

a time-varying ellipsoid which bounds the state trajectoryover the finite-time interval 119905 isin [0 119879] In addition wecan fix the upper bound beforehand in finite-time controland in general the initial and ultimates bound are not thesame Thus this concept will be more meaningful in prac-tice such as aircraft control networked control and robotoperating

4 Abstract and Applied Analysis

Definition 6 (see [38]) Let 119881(119909(119905) 119905) be the stochastic Lya-punov function of the stochastic system where 119909(119905) satisfiesthe following stochastic differential equation

119889119909 (119905) = 119891 (119905) 119889119905 + 119892 (119905) 119889119882 (119905) (11)

Define a weak infinitesimal operator L of random process119909(119905) 119905 gt 0

L119881 (119909 (119905) 119905) = 119881119905(119909 (119905) 119905) + 119881

119909(119909 (119905) 119905) 119891 (119905)

+1

2tr [119892Τ (119905) 119881

119909119909(119909 (119905) 119905) 119892 (119905)]

(12)

Lemma 7 (see [39] Gronwalls inequality) Let V(119905) be anonnegative function such that

V (119905) le 119886 + 119887int119905

0

V (119904) 119889119904 0 le 119905 le 119879 (13)

for some constants 119886 119887 ge 0 then we have

V (119905) le 119886 exp (119887119905) 0 le 119905 le 119879 (14)

Lemma 8 (Schurrsquos complement) For real matrices 119873119872119879 =119872 and 119877 = 119877

119879gt 0 the following three conditions are

equivalent

(1) 119872+119873119877minus1119873119879lt 0

(2) ( 119872 119873119873119879minus119877) lt 0

(3) ( 119872 minus119873minus119873119879minus119877) lt 0

Lemma 9 (see [40]) The following items are true

(i) Assume that rank(Ξ) = 119903 there exist two orthogonalmatrices119880 and119881 such that Ξ has the decomposition as

Ξ = 119880[Σ1199030

lowast 0]119881119879= 119880[

1198681199030

lowast 0] V119879 (15)

where Σ119903= diag120575

1 1205752 120575

119903 with 120575

119896gt 0 for all 119896 =

1 2 119903 Partition 119880 = [1198801 1198802] 119881 = [1198811 1198812] andV = [1198811Σ119903 1198812] with Ξ1198812 = 0 and 119880Τ2 Ξ = 0

(ii) If P satisfies

Ξ119875Τ= 119875ΞΤge 0 (16)

Then = 119880Τ119875V with119880 and V satisfying (8) if and onlyif

= [11987511

11987512

0 11987522

] (17)

with 11987511

ge 0 isin R119903times119903 In addition when P isnonsingular we have 119875

11gt 0 and det(119875

22) = 0

Furthermore 119875 satisfying (16) can be parameterized as

119875 = ΞVminusΤ119884Vminus1 + 119880119885119881Τ2 (18)

where 119884 = diag11987511 Ψ 119885 = [119875

Τ

12119875Τ

22]Τ

and Ψ isin

R(119899minus119903)times(119899minus119903) is an arbitrary parameter matrix

(iii) If P is a nonsingular matrix R and Ψ are two positivedefinite matrices P and Ξ satisfy (16) Y is a diagonalmatrix from (18) and the following equality holds

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (19)

Then the positive definite matrix 119876 = 119877minus12

119880119884minus1119880Τ

119877minus12 is a solution of (19)

3 FTS of the Stochastic Singular BiologicalEconomic System

This section provides finite-time stability analysis results forthe stochastic singular biological economic system somenovel sufficient conditions for finite-time stability of thestochastic singular system are given Through the previousanalysis we know that the system (2) and (3) are equivalentto the system (7) and (8) respectively Because the system (7)and (8) are regular the following theorems are given

31 FTS of the System (7)

Theorem 10 For a given time constant119879 gt 0 and a scalar 120572 gt0 the system (7) is finite-time stable with respect to (119888

1 1198882 119879 119877)

if there exist a nonsingular matrix 119875 and a symmetric positivedefinite matrix 119876 such that the following conditions hold

119875ΞΤ= Ξ119875Τge 0 (20)

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τlt 0 (21a)

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τlt 0 (21b)

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (22)

120582max (119876) 1198881119890120572119879minus 1198882120582min (119876) lt 0 (23)

Proof Consider the following Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (24)

We first prove that the following condition (25) is equivalentto (21a) and (21b)

(119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (25)

When 119905 isin [0 119879] we calculate the time derivative of119881(119883(119905) 119905)along the solution of system (7) and obtain

(119883 (119905) 119905) = 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎ)119883 (119905) (26)

which leads to

(119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ)119883 (119905)

(27)

Abstract and Applied Analysis 5

So (27) is equivalent to the following inequality

119860ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ lt 0 (28)

Pre- and postmultiplying (28) by 119875 and 119875Τ we get

119860ℎ119875Τ+ 119875119860Τ

ℎminus 120572Ξ119875

Τlt 0 (29)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decomposi-

tion we can see that (29) is equivalent to (21a) and (21b)Thus dividing by 119881(119883(119905) 119905) on both sides of (25) and

integrating it from 0 to 119905 with 119905 isin [0 119879] one gets

119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (30)

By condition (22) one can deduce that

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905)

= 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876)119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119909 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(31)

From (31) it is easy to obtain

119883Τ(119905) ΞΤ119877Ξ119883 (119905) le

120582max (119876)

120582min (119876)1198881119890120572119879 (32)

Considering condition (23) and inequality (32) it is easy toget 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes the

proof of the theorem

32 FTS of the System (8)

Theorem 11 For a given time constant 119879 gt 0 and a scalar120572 gt 0 the system (8) is stochastic finite-time stable with respectto (1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 and a

symmetric positive definite matrix 119876 such that (20) (22) and(23) hold and the following matrix inequalities are satisfied

[

[

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33a)

[

[

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33b)

where 119876 = 119877minus12

119876minus1119877minus12

Proof Let us consider the quadratic Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (34)

LetL be the infinitesimal generator (also called the averagedderivative) We first show the following condition

L119881 (119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (35)

Now we will prove that condition (35) is equivalent to (33a)and (33b) Applying Itorsquos formula it follows that

L119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862)119883 (119905)

(36)

which leads to

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ

+119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(37)

Combining condition (22) with 119876 = 119877minus12

119876minus1119877minus12 we can

obtain

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ

+119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(38)

So (37) is equivalent to

119860Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ lt 0 (39)

Pre- and postmultiplying (39) by 119875 and 119875Τ we get

119875119860Τ

ℎ+ 119860ℎ119875Τ+ 119875119862ΤΞΤ119876minus1Ξ119862119875Τminus 120572Ξ119875

Τlt 0 (40)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decompo-

sition and by Lemma 8 we can see that (40) is equivalent to(33a) and (33b)

Hence integrating both sides of (35) from 0 to 119905 with 119905 isin[0 119879] and then taking the expectation it yields

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) + 120572int

119905

0

120576 119881 (119883 (119904) 119904) 119889119904

(41)

By Lemma 7 we can obtain

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (42)

From (34) it follows that

120576 119881 (119883 (119905) 119905) = 120576 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876) 120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119883 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(43)

6 Abstract and Applied Analysis

From (43) it is easy to obtain

120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905) lt

120582max (119876)

120582min (119876)1198881119890120572119879 (44)

Considering condition (23) and inequality (44) it is easy toget 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes

the proof of the theorem

4 Finite-Time Control of the StochasticSingular Biological Economic System

Unstable fluctuations have always been regarded as unfa-vorable ones from the ecological managersrsquo point of viewMeanwhile price variation depends on the gap betweenthe demand function and supply function In order to planharvesting strategies and maintain the sustainable develop-ment of system it is necessary to take action to stabilizebiological population and market directed economy Thispaper proposes the following state feedback control methodfor the system (2) and (3)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(45)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

+ 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(46)

where 119906(119905) is control variable representing the managementof the open resource

Then we use the similar T-S fuzzy method shown inSection 2 and consider the following state feedback fuzzycontroller

119906 (119905) =

2

sum

119894=1

ℎ119894(119909 (119905)) 119871

119894119883 (119905) (47)

where 119871119894is the state feedback gain to be designed

Themodel (45) and (46) can be expressed by the followingfuzzy models

Ξ (119905) =

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905)) (119860

119894+ 119861119871119895)119883 (119905) (48)

Ξ119889119883 (119905)

=

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905))

times [(119860119894+ 119861119871119895)119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905)]

(49)

where 119861 = [1 0 0]Τ

In the sequence we provide finite-time stabilizationresults for above the mentioned closed-loop system (48) and(49) At the same time some novel sufficient conditions forfinite-time stabilization of the closed-loop singular biologicaleconomic system are given

41 Finite-Time Control of the System (45)

Theorem 12 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (48) is finite-time stable with respect to(1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 a symmetric

positive definite matrix 119876 and matrix 119884119895 119895 = 1 2 such

that (22) (23) hold and the following matrix inequalities aresatisfied

119875ΞΤ= Ξ119875Τge 0 (50)

Φ119894119894lt 0 119894 = 119895 119894 119895 = 1 2 (51a)

Φ119894119895+ Φ119895119894lt 0 119894 lt 119895 119894 119895 = 1 2 (51b)

where Φ119894119895= 119875119860Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ Then thedesired state feedback gain can be chosen as 119871

119895= 119884119895119875minusΤ

Proof Considering Theorem 10 and the closed-loop system(48) we can obtain the following matrix inequality

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) Π

119894119895lt 0 (52)

where Π119894119895= (119860119894+ 119861119871119895)119875Τ+ 119875(119860

119894+ 119861119871119895)Τminus 120572Ξ119875

ΤLet 119884119895= 119871119895119875Τ Then

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905))

times (119875119860Τ

119894+ 119884Τ

119895119861119879+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ) lt 0

(53)

Obviously (53) is equivalent to

2

sum

119894=1

ℎ2

119894(119909 (119905)) Φ

119894119894+

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) (Φ

119894119895+ Φ119895119894) lt 0

(54)

The above inequality (54) is equivalent to (51a) and (51b)Then by a similar argument in the proof of Theorem 10

and using (22)-(23) we can easily get the above results Thiscompletes the proof of the theorem

42 Finite-Time Control of the System (46)

Theorem 13 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (49) is stochastic finite-time stable withrespect to (119888

1 1198882 119879 119877) if there exist a nonsingular matrix 119875 a

symmetric positive definite matrix 119876 and matrix 119884119895 119895 = 1 2

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Definition 6 (see [38]) Let 119881(119909(119905) 119905) be the stochastic Lya-punov function of the stochastic system where 119909(119905) satisfiesthe following stochastic differential equation

119889119909 (119905) = 119891 (119905) 119889119905 + 119892 (119905) 119889119882 (119905) (11)

Define a weak infinitesimal operator L of random process119909(119905) 119905 gt 0

L119881 (119909 (119905) 119905) = 119881119905(119909 (119905) 119905) + 119881

119909(119909 (119905) 119905) 119891 (119905)

+1

2tr [119892Τ (119905) 119881

119909119909(119909 (119905) 119905) 119892 (119905)]

(12)

Lemma 7 (see [39] Gronwalls inequality) Let V(119905) be anonnegative function such that

V (119905) le 119886 + 119887int119905

0

V (119904) 119889119904 0 le 119905 le 119879 (13)

for some constants 119886 119887 ge 0 then we have

V (119905) le 119886 exp (119887119905) 0 le 119905 le 119879 (14)

Lemma 8 (Schurrsquos complement) For real matrices 119873119872119879 =119872 and 119877 = 119877

119879gt 0 the following three conditions are

equivalent

(1) 119872+119873119877minus1119873119879lt 0

(2) ( 119872 119873119873119879minus119877) lt 0

(3) ( 119872 minus119873minus119873119879minus119877) lt 0

Lemma 9 (see [40]) The following items are true

(i) Assume that rank(Ξ) = 119903 there exist two orthogonalmatrices119880 and119881 such that Ξ has the decomposition as

Ξ = 119880[Σ1199030

lowast 0]119881119879= 119880[

1198681199030

lowast 0] V119879 (15)

where Σ119903= diag120575

1 1205752 120575

119903 with 120575

119896gt 0 for all 119896 =

1 2 119903 Partition 119880 = [1198801 1198802] 119881 = [1198811 1198812] andV = [1198811Σ119903 1198812] with Ξ1198812 = 0 and 119880Τ2 Ξ = 0

(ii) If P satisfies

Ξ119875Τ= 119875ΞΤge 0 (16)

Then = 119880Τ119875V with119880 and V satisfying (8) if and onlyif

= [11987511

11987512

0 11987522

] (17)

with 11987511

ge 0 isin R119903times119903 In addition when P isnonsingular we have 119875

11gt 0 and det(119875

22) = 0

Furthermore 119875 satisfying (16) can be parameterized as

119875 = ΞVminusΤ119884Vminus1 + 119880119885119881Τ2 (18)

where 119884 = diag11987511 Ψ 119885 = [119875

Τ

12119875Τ

22]Τ

and Ψ isin

R(119899minus119903)times(119899minus119903) is an arbitrary parameter matrix

(iii) If P is a nonsingular matrix R and Ψ are two positivedefinite matrices P and Ξ satisfy (16) Y is a diagonalmatrix from (18) and the following equality holds

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (19)

Then the positive definite matrix 119876 = 119877minus12

119880119884minus1119880Τ

119877minus12 is a solution of (19)

3 FTS of the Stochastic Singular BiologicalEconomic System

This section provides finite-time stability analysis results forthe stochastic singular biological economic system somenovel sufficient conditions for finite-time stability of thestochastic singular system are given Through the previousanalysis we know that the system (2) and (3) are equivalentto the system (7) and (8) respectively Because the system (7)and (8) are regular the following theorems are given

31 FTS of the System (7)

Theorem 10 For a given time constant119879 gt 0 and a scalar 120572 gt0 the system (7) is finite-time stable with respect to (119888

1 1198882 119879 119877)

if there exist a nonsingular matrix 119875 and a symmetric positivedefinite matrix 119876 such that the following conditions hold

119875ΞΤ= Ξ119875Τge 0 (20)

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τlt 0 (21a)

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τlt 0 (21b)

119875minus1Ξ = Ξ

Τ1198771211987611987712Ξ (22)

120582max (119876) 1198881119890120572119879minus 1198882120582min (119876) lt 0 (23)

Proof Consider the following Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (24)

We first prove that the following condition (25) is equivalentto (21a) and (21b)

(119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (25)

When 119905 isin [0 119879] we calculate the time derivative of119881(119883(119905) 119905)along the solution of system (7) and obtain

(119883 (119905) 119905) = 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎ)119883 (119905) (26)

which leads to

(119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ)119883 (119905)

(27)

Abstract and Applied Analysis 5

So (27) is equivalent to the following inequality

119860ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ lt 0 (28)

Pre- and postmultiplying (28) by 119875 and 119875Τ we get

119860ℎ119875Τ+ 119875119860Τ

ℎminus 120572Ξ119875

Τlt 0 (29)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decomposi-

tion we can see that (29) is equivalent to (21a) and (21b)Thus dividing by 119881(119883(119905) 119905) on both sides of (25) and

integrating it from 0 to 119905 with 119905 isin [0 119879] one gets

119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (30)

By condition (22) one can deduce that

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905)

= 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876)119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119909 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(31)

From (31) it is easy to obtain

119883Τ(119905) ΞΤ119877Ξ119883 (119905) le

120582max (119876)

120582min (119876)1198881119890120572119879 (32)

Considering condition (23) and inequality (32) it is easy toget 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes the

proof of the theorem

32 FTS of the System (8)

Theorem 11 For a given time constant 119879 gt 0 and a scalar120572 gt 0 the system (8) is stochastic finite-time stable with respectto (1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 and a

symmetric positive definite matrix 119876 such that (20) (22) and(23) hold and the following matrix inequalities are satisfied

[

[

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33a)

[

[

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33b)

where 119876 = 119877minus12

119876minus1119877minus12

Proof Let us consider the quadratic Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (34)

LetL be the infinitesimal generator (also called the averagedderivative) We first show the following condition

L119881 (119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (35)

Now we will prove that condition (35) is equivalent to (33a)and (33b) Applying Itorsquos formula it follows that

L119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862)119883 (119905)

(36)

which leads to

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ

+119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(37)

Combining condition (22) with 119876 = 119877minus12

119876minus1119877minus12 we can

obtain

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ

+119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(38)

So (37) is equivalent to

119860Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ lt 0 (39)

Pre- and postmultiplying (39) by 119875 and 119875Τ we get

119875119860Τ

ℎ+ 119860ℎ119875Τ+ 119875119862ΤΞΤ119876minus1Ξ119862119875Τminus 120572Ξ119875

Τlt 0 (40)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decompo-

sition and by Lemma 8 we can see that (40) is equivalent to(33a) and (33b)

Hence integrating both sides of (35) from 0 to 119905 with 119905 isin[0 119879] and then taking the expectation it yields

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) + 120572int

119905

0

120576 119881 (119883 (119904) 119904) 119889119904

(41)

By Lemma 7 we can obtain

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (42)

From (34) it follows that

120576 119881 (119883 (119905) 119905) = 120576 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876) 120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119883 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(43)

6 Abstract and Applied Analysis

From (43) it is easy to obtain

120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905) lt

120582max (119876)

120582min (119876)1198881119890120572119879 (44)

Considering condition (23) and inequality (44) it is easy toget 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes

the proof of the theorem

4 Finite-Time Control of the StochasticSingular Biological Economic System

Unstable fluctuations have always been regarded as unfa-vorable ones from the ecological managersrsquo point of viewMeanwhile price variation depends on the gap betweenthe demand function and supply function In order to planharvesting strategies and maintain the sustainable develop-ment of system it is necessary to take action to stabilizebiological population and market directed economy Thispaper proposes the following state feedback control methodfor the system (2) and (3)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(45)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

+ 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(46)

where 119906(119905) is control variable representing the managementof the open resource

Then we use the similar T-S fuzzy method shown inSection 2 and consider the following state feedback fuzzycontroller

119906 (119905) =

2

sum

119894=1

ℎ119894(119909 (119905)) 119871

119894119883 (119905) (47)

where 119871119894is the state feedback gain to be designed

Themodel (45) and (46) can be expressed by the followingfuzzy models

Ξ (119905) =

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905)) (119860

119894+ 119861119871119895)119883 (119905) (48)

Ξ119889119883 (119905)

=

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905))

times [(119860119894+ 119861119871119895)119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905)]

(49)

where 119861 = [1 0 0]Τ

In the sequence we provide finite-time stabilizationresults for above the mentioned closed-loop system (48) and(49) At the same time some novel sufficient conditions forfinite-time stabilization of the closed-loop singular biologicaleconomic system are given

41 Finite-Time Control of the System (45)

Theorem 12 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (48) is finite-time stable with respect to(1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 a symmetric

positive definite matrix 119876 and matrix 119884119895 119895 = 1 2 such

that (22) (23) hold and the following matrix inequalities aresatisfied

119875ΞΤ= Ξ119875Τge 0 (50)

Φ119894119894lt 0 119894 = 119895 119894 119895 = 1 2 (51a)

Φ119894119895+ Φ119895119894lt 0 119894 lt 119895 119894 119895 = 1 2 (51b)

where Φ119894119895= 119875119860Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ Then thedesired state feedback gain can be chosen as 119871

119895= 119884119895119875minusΤ

Proof Considering Theorem 10 and the closed-loop system(48) we can obtain the following matrix inequality

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) Π

119894119895lt 0 (52)

where Π119894119895= (119860119894+ 119861119871119895)119875Τ+ 119875(119860

119894+ 119861119871119895)Τminus 120572Ξ119875

ΤLet 119884119895= 119871119895119875Τ Then

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905))

times (119875119860Τ

119894+ 119884Τ

119895119861119879+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ) lt 0

(53)

Obviously (53) is equivalent to

2

sum

119894=1

ℎ2

119894(119909 (119905)) Φ

119894119894+

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) (Φ

119894119895+ Φ119895119894) lt 0

(54)

The above inequality (54) is equivalent to (51a) and (51b)Then by a similar argument in the proof of Theorem 10

and using (22)-(23) we can easily get the above results Thiscompletes the proof of the theorem

42 Finite-Time Control of the System (46)

Theorem 13 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (49) is stochastic finite-time stable withrespect to (119888

1 1198882 119879 119877) if there exist a nonsingular matrix 119875 a

symmetric positive definite matrix 119876 and matrix 119884119895 119895 = 1 2

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

So (27) is equivalent to the following inequality

119860ℎ

Τ119875minusΤ+ 119875minus1119860ℎminus 120572119875minus1Ξ lt 0 (28)

Pre- and postmultiplying (28) by 119875 and 119875Τ we get

119860ℎ119875Τ+ 119875119860Τ

ℎminus 120572Ξ119875

Τlt 0 (29)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decomposi-

tion we can see that (29) is equivalent to (21a) and (21b)Thus dividing by 119881(119883(119905) 119905) on both sides of (25) and

integrating it from 0 to 119905 with 119905 isin [0 119879] one gets

119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (30)

By condition (22) one can deduce that

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905)

= 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876)119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119909 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(31)

From (31) it is easy to obtain

119883Τ(119905) ΞΤ119877Ξ119883 (119905) le

120582max (119876)

120582min (119876)1198881119890120572119879 (32)

Considering condition (23) and inequality (32) it is easy toget 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes the

proof of the theorem

32 FTS of the System (8)

Theorem 11 For a given time constant 119879 gt 0 and a scalar120572 gt 0 the system (8) is stochastic finite-time stable with respectto (1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 and a

symmetric positive definite matrix 119876 such that (20) (22) and(23) hold and the following matrix inequalities are satisfied

[

[

1198601119875Τ+ 119875119860Τ

1minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33a)

[

[

1198602119875Τ+ 119875119860Τ

2minus 120572Ξ119875

Τ(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 (33b)

where 119876 = 119877minus12

119876minus1119877minus12

Proof Let us consider the quadratic Lyapunov function

119881 (119883 (119905) 119905) = 119883Τ(119905) 119875minus1Ξ119883 (119905) (34)

LetL be the infinitesimal generator (also called the averagedderivative) We first show the following condition

L119881 (119883 (119905) 119905) lt 120572119881 (119883 (119905) 119905) (35)

Now we will prove that condition (35) is equivalent to (33a)and (33b) Applying Itorsquos formula it follows that

L119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862)119883 (119905)

(36)

which leads to

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ

+119875minus1119860ℎ+ 119862Τ119875minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(37)

Combining condition (22) with 119876 = 119877minus12

119876minus1119877minus12 we can

obtain

L119881 (119883 (119905) 119905) minus 120572119881 (119883 (119905) 119905)

= 119883Τ(119905) (119860

Τ

ℎ119875minusΤ+ 119875minus1119860ℎ

+119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ)119883 (119905)

(38)

So (37) is equivalent to

119860Τ

ℎ119875minusΤ+ 119875minus1119860ℎ+ 119862ΤΞΤ119876minus1Ξ119862 minus 120572119875

minus1Ξ lt 0 (39)

Pre- and postmultiplying (39) by 119875 and 119875Τ we get

119875119860Τ

ℎ+ 119860ℎ119875Τ+ 119875119862ΤΞΤ119876minus1Ξ119862119875Τminus 120572Ξ119875

Τlt 0 (40)

When 119860ℎ= sum2

119894=1ℎ119894(119909(119905))119860

119894 through the matrix decompo-

sition and by Lemma 8 we can see that (40) is equivalent to(33a) and (33b)

Hence integrating both sides of (35) from 0 to 119905 with 119905 isin[0 119879] and then taking the expectation it yields

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) + 120572int

119905

0

120576 119881 (119883 (119904) 119904) 119889119904

(41)

By Lemma 7 we can obtain

120576 119881 (119883 (119905) 119905) lt 119881 (119883 (0) 0) 119890120572119905 (42)

From (34) it follows that

120576 119881 (119883 (119905) 119905) = 120576 119883Τ(119905) ΞΤ1198771211987611987712Ξ119883 (119905)

ge 120582min (119876) 120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905)

119881 (119883 (0) 0) 119890120572119905= 119883Τ(0) ΞΤ1198771211987611987712Ξ119883 (0) 119890

120572119905

le 120582max (119876)119883Τ(0) ΞΤ119877Ξ119883 (0) 119890

120572119905

le 120582max (119876) 1198881119890120572119879

(43)

6 Abstract and Applied Analysis

From (43) it is easy to obtain

120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905) lt

120582max (119876)

120582min (119876)1198881119890120572119879 (44)

Considering condition (23) and inequality (44) it is easy toget 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes

the proof of the theorem

4 Finite-Time Control of the StochasticSingular Biological Economic System

Unstable fluctuations have always been regarded as unfa-vorable ones from the ecological managersrsquo point of viewMeanwhile price variation depends on the gap betweenthe demand function and supply function In order to planharvesting strategies and maintain the sustainable develop-ment of system it is necessary to take action to stabilizebiological population and market directed economy Thispaper proposes the following state feedback control methodfor the system (2) and (3)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(45)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

+ 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(46)

where 119906(119905) is control variable representing the managementof the open resource

Then we use the similar T-S fuzzy method shown inSection 2 and consider the following state feedback fuzzycontroller

119906 (119905) =

2

sum

119894=1

ℎ119894(119909 (119905)) 119871

119894119883 (119905) (47)

where 119871119894is the state feedback gain to be designed

Themodel (45) and (46) can be expressed by the followingfuzzy models

Ξ (119905) =

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905)) (119860

119894+ 119861119871119895)119883 (119905) (48)

Ξ119889119883 (119905)

=

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905))

times [(119860119894+ 119861119871119895)119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905)]

(49)

where 119861 = [1 0 0]Τ

In the sequence we provide finite-time stabilizationresults for above the mentioned closed-loop system (48) and(49) At the same time some novel sufficient conditions forfinite-time stabilization of the closed-loop singular biologicaleconomic system are given

41 Finite-Time Control of the System (45)

Theorem 12 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (48) is finite-time stable with respect to(1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 a symmetric

positive definite matrix 119876 and matrix 119884119895 119895 = 1 2 such

that (22) (23) hold and the following matrix inequalities aresatisfied

119875ΞΤ= Ξ119875Τge 0 (50)

Φ119894119894lt 0 119894 = 119895 119894 119895 = 1 2 (51a)

Φ119894119895+ Φ119895119894lt 0 119894 lt 119895 119894 119895 = 1 2 (51b)

where Φ119894119895= 119875119860Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ Then thedesired state feedback gain can be chosen as 119871

119895= 119884119895119875minusΤ

Proof Considering Theorem 10 and the closed-loop system(48) we can obtain the following matrix inequality

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) Π

119894119895lt 0 (52)

where Π119894119895= (119860119894+ 119861119871119895)119875Τ+ 119875(119860

119894+ 119861119871119895)Τminus 120572Ξ119875

ΤLet 119884119895= 119871119895119875Τ Then

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905))

times (119875119860Τ

119894+ 119884Τ

119895119861119879+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ) lt 0

(53)

Obviously (53) is equivalent to

2

sum

119894=1

ℎ2

119894(119909 (119905)) Φ

119894119894+

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) (Φ

119894119895+ Φ119895119894) lt 0

(54)

The above inequality (54) is equivalent to (51a) and (51b)Then by a similar argument in the proof of Theorem 10

and using (22)-(23) we can easily get the above results Thiscompletes the proof of the theorem

42 Finite-Time Control of the System (46)

Theorem 13 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (49) is stochastic finite-time stable withrespect to (119888

1 1198882 119879 119877) if there exist a nonsingular matrix 119875 a

symmetric positive definite matrix 119876 and matrix 119884119895 119895 = 1 2

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

From (43) it is easy to obtain

120576 119883Τ(119905) ΞΤ119877Ξ119883 (119905) lt

120582max (119876)

120582min (119876)1198881119890120572119879 (44)

Considering condition (23) and inequality (44) it is easy toget 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 119888

2for all 119905 isin [0 119879] This completes

the proof of the theorem

4 Finite-Time Control of the StochasticSingular Biological Economic System

Unstable fluctuations have always been regarded as unfa-vorable ones from the ecological managersrsquo point of viewMeanwhile price variation depends on the gap betweenthe demand function and supply function In order to planharvesting strategies and maintain the sustainable develop-ment of system it is necessary to take action to stabilizebiological population and market directed economy Thispaper proposes the following state feedback control methodfor the system (2) and (3)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(45)

119889119909 (119905) = [119903119909 (119905) (1 minus119909 (119905)

119896) minus 119864119909 (119905) + 119906 (119905)] 119889119905

+ 120590119909 (119905) 119889119882 (119905)

119889119901 (119905) = 119904 (119886 minus 119901 (119905) minus 119864119909 (119905)) 119889119905

0 = 119864119901 (119905) 119909 (119905) minus 119888119864119909 (119905) minus 119898 (119905)

(46)

where 119906(119905) is control variable representing the managementof the open resource

Then we use the similar T-S fuzzy method shown inSection 2 and consider the following state feedback fuzzycontroller

119906 (119905) =

2

sum

119894=1

ℎ119894(119909 (119905)) 119871

119894119883 (119905) (47)

where 119871119894is the state feedback gain to be designed

Themodel (45) and (46) can be expressed by the followingfuzzy models

Ξ (119905) =

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905)) (119860

119894+ 119861119871119895)119883 (119905) (48)

Ξ119889119883 (119905)

=

2

sum

119894=1

ℎ119894(119909 (119905))

2

sum

119895=1

ℎ119895(119909 (119905))

times [(119860119894+ 119861119871119895)119883 (119905) 119889119905 + 119862119883 (119905) 119889119882 (119905)]

(49)

where 119861 = [1 0 0]Τ

In the sequence we provide finite-time stabilizationresults for above the mentioned closed-loop system (48) and(49) At the same time some novel sufficient conditions forfinite-time stabilization of the closed-loop singular biologicaleconomic system are given

41 Finite-Time Control of the System (45)

Theorem 12 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (48) is finite-time stable with respect to(1198881 1198882 119879 119877) if there exist a nonsingular matrix 119875 a symmetric

positive definite matrix 119876 and matrix 119884119895 119895 = 1 2 such

that (22) (23) hold and the following matrix inequalities aresatisfied

119875ΞΤ= Ξ119875Τge 0 (50)

Φ119894119894lt 0 119894 = 119895 119894 119895 = 1 2 (51a)

Φ119894119895+ Φ119895119894lt 0 119894 lt 119895 119894 119895 = 1 2 (51b)

where Φ119894119895= 119875119860Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ Then thedesired state feedback gain can be chosen as 119871

119895= 119884119895119875minusΤ

Proof Considering Theorem 10 and the closed-loop system(48) we can obtain the following matrix inequality

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) Π

119894119895lt 0 (52)

where Π119894119895= (119860119894+ 119861119871119895)119875Τ+ 119875(119860

119894+ 119861119871119895)Τminus 120572Ξ119875

ΤLet 119884119895= 119871119895119875Τ Then

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905))

times (119875119860Τ

119894+ 119884Τ

119895119861119879+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ) lt 0

(53)

Obviously (53) is equivalent to

2

sum

119894=1

ℎ2

119894(119909 (119905)) Φ

119894119894+

2

sum

119894=1

2

sum

119895=1

ℎ119894(119909 (119905)) ℎ

119895(119909 (119905)) (Φ

119894119895+ Φ119895119894) lt 0

(54)

The above inequality (54) is equivalent to (51a) and (51b)Then by a similar argument in the proof of Theorem 10

and using (22)-(23) we can easily get the above results Thiscompletes the proof of the theorem

42 Finite-Time Control of the System (46)

Theorem 13 For a given time constant 119879 gt 0 and a scalar120572 gt 0 there exists a state feedback controller (47) such thatthe closed-loop system (49) is stochastic finite-time stable withrespect to (119888

1 1198882 119879 119877) if there exist a nonsingular matrix 119875 a

symmetric positive definite matrix 119876 and matrix 119884119895 119895 = 1 2

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

such that (50) (22) and (23) hold and the following matrixinequalities are satisfied

[

[

Ψ119894119894

(Ξ119862119875Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 = 119895 119894 119895 = 1 2 (55a)

[

[

Ψ119894119895+ Ψ119895119894(Ξ119862119875

Τ)Τ

Ξ119862119875Τ

minus119876

]

]

lt 0 119894 lt 119895 119894 119895 = 1 2 (55b)

where Ψ119894119895= 119875119860

Τ

119894+ 119884Τ

119895119861Τ+ 119860119894119875Τ+ 119861119884119895minus 120572Ξ119875

Τ and 119876 =

119877minus12

119876minus1119877minus12 Then the desired state feedback gain can be

chosen as 119871119895= 119884119895119875minusΤ

Proof Considering Theorem 11 and the closed-loop system(49) this proof follows a similar way with the proof inTheorem 12 and it is easy to obtain the previous results Thiscompletes the proof of the theorem

Remark 14 The state feedback control method can be imple-mented by the management of the open resource so thatthe biological population can be controlled at the boundedrange and those unfavorable phenomena will be eliminatedIn our daily life managers can take action such as adjustingrevenue drawing out favorable policy to encourage fisheryand abating pollution in order to control the populationdensity and economic profit

5 Numerical Examples

To illustrate the results obtained let us consider the follow-ing particular cases We choose the ecological parametersaccording to biological sardine data of theMoroccan westerncoastline area and these parameters are given in [6]

Example 1 Consider finite-time stabilization problem ofthe biological economic model and select the ecologicalparameters as 119903 = 0015 119896 = 17028 119864 = 0008 119904 = 0005and 119888 = 200 119886 = 800 in appropriate units Then the closed-loop system (48) can be described with the following data

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119861 = [1 0 0]Τ

(56)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 119877 = 119868

By Theorem 12 we can calculate that

119875 = 105times [

[

00145 00003 00463

00003 00002 00016

0 0 18983

]

]

119876 = [

[

00007 minus00014 0

minus00014 00580 0

0 0 00294

]

]

1198711= [minus366991 692352 minus00098]

1198712= [minus366538 692506 minus00147]

(57)

So we claim that 119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 forall119905 isin

[0 1000] The trajectory is shown in Figure 2Figure 1 shows the state trajectory of the open-loop

biological economic system when 119905 isin [0 1000] Figure 2shows the state trajectory of the closed-loop biological eco-nomic system with state feedback controller (47) when 119905 isin[0 1000]

Example 2 Consider finite-time stabilization problem of thebiological economicmodel with thewhite noise in Example 11205902 is intensities of the white noise and let 1205902 = 001 Select

the ecological parameters as 119903 = 0015 119896 = 17028119864 = 0008119904 = 0005 119888 = 200 and 119886 = 800 in appropriate units Theclosed-loop system (49) can be formulated with the followingparameters

Ξ = [

[

1 0 0

0 1 0

0 0 0

]

]

1198601= [

[

00220 00000 00000

minus00000 minus00050 00000

48000 minus136224 minus10000

]

]

119861 = [1 0 0]Τ

1198602= [

[

minus00080 00000 00000

minus00000 minus00050 00000

48000 136224 minus10000

]

]

119862 = [

[

01 0 0

0 0 0

0 0 0

]

]

(58)

Let 120572 = 0001 1198881= 10000 119888

2= 10000000 119879 = 1000 and

119877 = 119868 By Theorem 13 we can calculate that

119875 = 105times [

[

00837 00020 02679

00020 00004 00098

0 0 37772

]

]

119876 = [

[

00001 minus00007 0

minus00007 00302 0

0 0 00049

]

]

1198711= [minus86579 472591 minus00345]

1198712= [minus86221 472785 minus00364]

(59)

So we claim that 120576119883Τ(119905)ΞΤ119877Ξ119883(119905) lt 10000000 for all 119905 isin[0 1000] The trajectory is shown in Figure 4

Figure 3 shows the state trajectory of the open-loopstochastic singular biological economic systemwith thewhitenoise 1205902 Taking 1205902 = 001 we know that the populationdensity and the unit price are not stable over the finite-timeinterval [0 1000] The economic profit is random fluctuationin Figure 3 because the economic profit is affected by theunit price of the population and the population density ofour proposed biological economic model in the randomdisturbance environment

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 1 Trajectory of the open-loop biological economic system

0 200 400 600 800 10000

2000

4000

6000

8000

10000

12000

14000

16000

18000

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

Figure 2 Trajectory of the closed-loop biological economic system

Figure 4 shows the state trajectory of the closed-loopstochastic singular biological economic systemwith thewhitenoise 1205902 = 001 through the state feedback controller (47)From Figure 4 it can be seen that the economic profit tendsto be stable over the finite-time interval [0 1000]

6 Conclusions

In this paper we study the problems of finite-time stabilityand control for a class of stochastic singular biologicaleconomic systems based onT-S fuzzymodel By the theorems

0 200 400 600 800 10000

500

1000

1500

2000

2500

3000

3500

4000

Time

Stat

e res

pons

e of t

he o

pen-

loop

syste

m

x(t)

p(t)

m(t)

Figure 3 Trajectory of the open-loop stochastic singular system

0 100 200 300 400 500 600 700 800 900 10000

02

04

06

08

1

12

14

16

18

2

Time

Stat

e res

pons

e of t

he cl

osed

-loop

syste

m

x(t)

p(t)

m(t)

times104

Figure 4 Trajectory of the closed-loop stochastic singular system

we presented some novel sufficient conditions are derivedwhich ensure that the systems are stable in finite timeAnd then the corresponding controller design methods areobtained Finally numerical examples are presented to illus-trate the effectiveness of the proposed results Our proposedresults in this paper can also be applied to the study of otherclasses of systems

Acknowledgment

This work was supported by National Natural Science Foun-dation of China underGrant no 60974004 and no 61273008

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

References

[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resources Pure and Applied MathematicsJohn Wiley amp Sons New York NY USA 2nd edition 1990

[2] M Jerry and N Raıssi ldquoA policy of fisheries management basedon continuous fishing effortrdquo Journal of Biological Systems vol9 no 4 pp 247ndash254 2001

[3] M Jerry and N Raıssi ldquoThe optimal strategy for a bioeco-nomical model of a harvesting renewable resource problemrdquoMathematical and Computer Modelling vol 36 no 11-13 pp1293ndash1306 2002

[4] R Q Grafton L K Sandal and S I Steinshamn ldquoHow toimprove the management of renewable resources the case ofCanadarsquos northern cod fisheryrdquoAmerican Journal of AgriculturalEconomics vol 82 no 3 pp 570ndash580 2000

[5] M B Schaefer ldquoSome aspects of the dynamics of populationsimportant to the management of the commercial marinefisheriesrdquo Bulletin of Mathematical Biology vol 53 no 1-2 pp253ndash279 1991

[6] C Jerry and N Raıssi ldquoCan management measures ensure thebiological and economical stabilizability of a fishing modelrdquoMathematical and Computer Modelling vol 51 no 5-6 pp 516ndash526 2010

[7] C Bernstein P Auger and J C Poggiale ldquoPredator migrationdecisions the ideal free distribution and predator-prey dynam-icsrdquo American Naturalist vol 153 no 3 pp 267ndash281 1999

[8] J T Lafrance ldquoLinear demand functions in theory and practicerdquoJournal of Economic Theory vol 37 no 1 pp 147ndash166 1985

[9] CMullon andP Freon ldquoPrototype of an integratedmodel of theworldwide system of small pelagic fisheriesrdquo in Climate Changeand the Economics of the Worldrsquos Fisheries Examples of SmallPelagic Stocks New Horizons in Environmental Economics pp262ndash295 Edward Elgar Cheltenham UK 2005

[10] L Lasselle S Svizzero and C Tisdell ldquoStability and cycles ina cobweb model with heterogeneous expectationsrdquo Macroeco-nomic Dynamics vol 9 no 5 pp 630ndash650 2005

[11] S Ayasun C O Nwankpa and H G Kwatny ldquoComputa-tion of singular and singularity induced bifurcation points ofdifferential-algebraic power system modelrdquo IEEE TransactionsonCircuits and Systems I Regular Papers vol 51 no 8 pp 1525ndash1538 2004

[12] R Riaza ldquoSingularity-induced bifurcations in lumped circuitsrdquoIEEE Transactions on Circuits and Systems I vol 52 no 7 pp1442ndash1450 2005

[13] D G Luenberger ldquoNonlinear descriptor systemsrdquo Journal ofEconomic Dynamics amp Control vol 1 no 3 pp 219ndash242 1979

[14] H Krishnan and N H McClamroch ldquoTracking in nonlineardifferential-algebraic control systems with applications to con-strained robot systemsrdquo Automatica vol 30 no 12 pp 1885ndash1897 1994

[15] A M Bloch M Reyhanoglu and N H McClamroch ldquoControland stabilization of nonholonomic dynamic systemsrdquo IEEETransactions on Automatic Control vol 37 no 11 pp 1746ndash17571992

[16] H S Gordon ldquoEconomic theory of a common propertyresource the fisheryrdquo The Journal of Political Economy vol 62no 2 pp 124ndash142 1954

[17] X Zhang Q-l Zhang and Y Zhang ldquoBifurcations of a classof singular biological economic modelsrdquo Chaos Solitons andFractals vol 40 no 3 pp 1309ndash1318 2009

[18] C Liu Q Zhang Y Zhang and X Duan ldquoBifurcation and con-trol in a differential-algebraic harvested prey-predator modelwith stage structure for predatorrdquo International Journal ofBifurcation and Chaos vol 18 no 10 pp 3159ndash3168 2008

[19] Y Zhang and Q L Zhang ldquoBifurcations and control in singularbiological economic model with stage structurerdquo Journal ofSystems Engineering vol 22 no 3 pp 233ndash238 2007

[20] Q Zhang C Liu and X Zhang Complexity Analysis andControl of Singular Biological Systems vol 421 of LectureNotes inControl and Information Sciences Springer London UK 2012

[21] R M May Stability and Complexity in Model EcosystemsPrinceton University Press NJ USA 1973

[22] M Carletti ldquoNumerical simulation of a Campbell-like stochas-tic delay model for bacteriophage infectionrdquo MathematicalMedicine and Biology vol 23 no 4 pp 297ndash310 2006

[23] X Mao G Marion and E Renshaw ldquoEnvironmental Browniannoise suppresses explosions in population dynamicsrdquo StochasticProcesses and their Applications vol 97 no 1 pp 95ndash110 2002

[24] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[25] C Y Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

[26] C Ji and D Jiang ldquoDynamics of a stochastic density dependentpredator-prey system with Beddington-DeAngelis functionalresponserdquo Journal of Mathematical Analysis and Applicationsvol 381 no 1 pp 441ndash453 2011

[27] J Lv and K Wang ldquoAsymptotic properties of a stochasticpredator-prey system with Holling II functional responserdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 10 pp 4037ndash4048 2011

[28] Z W Liu N Z Shi D Q Jiang and C Y Ji ldquoThe asymptoticbehavior of a stochastic predator-prey system with holling IIfunctional responserdquo Abstract and Applied Analysis vol 2012Article ID 801812 14 pages 2012

[29] X Song H Zhang and L Xie ldquoStochastic linear quadraticregulation for discrete-time linear systems with input delayrdquoAutomatica vol 45 no 9 pp 2067ndash2073 2009

[30] B-S Chen and W Zhang ldquoStochastic 1198672119867infin

control withstate-dependent noiserdquo IEEE Transactions on Automatic Con-trol vol 49 no 1 pp 45ndash57 2004

[31] P Doroto ldquoShort time stability in linear time-varying systemsrdquoProceeding of IRE International Convention Record vol 4 pp83ndash87 1961

[32] F Amato M Ariola and P Dorato ldquoFinite-time control oflinear systems subject to parametric uncertainties and distur-bancesrdquo Automatica vol 37 no 9 pp 1459ndash1463 2001

[33] F Amato and M Ariola ldquoFinite-time control of discrete-timelinear systemsrdquo IEEETransactions onAutomatic Control vol 50no 5 pp 724ndash729 2005

[34] L Liu and J Sun ldquoFinite-time stabilization of linear systems viaimpulsive controlrdquo International Journal of Control vol 81 no6 pp 905ndash909 2008

[35] S Zhao J Sun and L Liu ldquoFinite-time stability of linear time-varying singular systems with impulsive effectsrdquo InternationalJournal of Control vol 81 no 11 pp 1824ndash1829 2008

[36] W Zhang and X An ldquoFinite-time control of linear stochasticsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 4 no 3 pp 689ndash696 2008

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[37] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[38] S G Hu B M Huang and F K Wu Stochastic DifferentialEquations Science Press Beijing China 2008

[39] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer New York NY USA 5th edition2000

[40] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of

singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of