matrix measure strategies for stability of cellular neural networks with proportional ... · 2016....

Matrix Measure Strategies for Stability of CellularNeural Networks with Proportional Delays

Changjin Xu, and Peiluan Li

Abstract—This paper is concerned with cellular neural net-works with proportional delays. The proportional delay is atime-varying unbounded delay which is different from the con-stant delay, bounded time-varying delay and distributed delay.Using matrix measure and generalized Halanay inequality, somesufficient conditions are obtained to ensure the pth exponentialstability of cellular neural networks with proportional delays.The obtained results are simple and easy to be verified. Anexample is given to illustrate the effectiveness of the obtainedresults. This paper ends with a brief conclusion.

Index Terms—Cellular neural networks, exponential stability,proportional delays, matrix measure.

I. INTRODUCTION

IT is well known that considerable attention has been paidto cellular neural networks as well as various general-izations for their potential applications in many fields suchas associative memories, pattern recognition, optimizationand image processing and so on [1-13]. On the one hand,the existence and stability of the equilibrium point of cel-lular neural networks plays an important role in practicalapplication. On the other hand, time delay is inevitable dueto the finite switching speed of information processing andthe inherent communication time of neurons, moreover, itsexistence may cause the instability of networks [14]. Thusmany interesting stability results on cellular neural networkswith delays have been available [14-19]. At present, thetime delays considered for cellular neural networks can beclassified as constant delays [4,16,21], time-varying delays[3,15,19,21], and distributed delays [22-24]. Here we wouldlike to point out that the proportional delay which is a specialdelay type exists in some fields such as physics, biologysystems , control theory and Web quality of service (QoS)routing decision. Since the presence of an amount of parallelpathways of a variety of axon sizes and lengths, a neuralnetwork usually has a spatial structure, it is reasonable tointroduce the proportional delays into the neural networks.In an amount of parallel pathways, affected by differentmaterials and topology, there may be some unbounded delaysthat is proportional to the time, thus we should choosesuitable proportional delays factors in view of different

Manuscript received December 21, 2015; revised March 2, 2016. Thiswork was supported in part by the This work is supported by National Nat-ural Science Foundation of China(No.11261010 and No.11101126), NaturalScience and Technology Foundation of Guizhou Province(J[2015]2025 andJ[2015]2026), 125 Special Major Science and Technology of Department ofEducation of Guizhou Province ([2012]011) and Natural Science Foundationof the Education Department of Guizhou Province(KY[2015]482). E-mail:[email protected]

C. Xu is with the Guizhou Key Laboratory of Economics System Sim-ulation, Guizhou University of Finance and Economics, Guiyang 550004,PR China e-mail: [email protected].

P. Li is with School of Mathematics and Statistics, Henan Univer-sity of Science and Technology, Luoyang 471023, PR China e-mail:lpllpl−[email protected]

cases and adopt proportional delays to characterize theseunbounded delays [25]. Recently, there are only very fewpapers that focus on this aspect. For example, Zhou et al.[14] considered the asymptotic stability of cellular neuralnetworks with multiple proportional delays, Zheng et al. [26]established the stability criteria for high-order networks withproportional delay. Zhou [27] adressed the delay-dependentexponential stability of cellular neural networks with multi-proportional delays, Zhou [28] discussed the delay-dependentexponential synchronization of recurrent neural works withmultiple proportional delays, Zhou [29] analyzed the globalasymptotic stability of cellular neural networks with propor-tional delays, Zhou [30] investigated the dissipativity of aclass of cellular neural networks with proportional delays.In details, one can see [34,35]. We must point out thatcellular neural networks with proportional delays have beenwidely applied in many fields such as light absorption inthe star substance and nonlinear dynamic systems. Thereforethe study on the cellular neural networks with proportionaldelays has important theoretical and practical value.

In 2015, Zhou and Zhang [31] investigated the globalexponential stability of the following cellular neural networkswith multi-proportional delays

ẋi(t) = −dixi(t) +n∑

j=1

[aijfj(xj(t)) + bijgj(xj(q1t))

+ cijhj(xj(q2t))] + Ii,xi(s) = xi0, s ∈ [q, 1],

(1)where i = 1, 2, · · · , n, t > 1, xi(t) stands for the statevariable of the ith-cell at time t, di > 0 is a constant, aij , bijand cij represent the connection weights between the ith-celland the jth-cell at time t, q1t, q2t, respectively. q1 and q2 areproportional delay factors and satisfy 0 < q1 < q2 ≤ 1, q =min{q1, q2}, q1t = t − (1 − q1)t, q2t = t − (1 − q2t)t inwhich (1−q1)t and (1−q2)t denote the transmission delays,(1 − q1)t → +∞, (1 − q2)t → +∞ as t → +∞, xi0 is aconstant which denotes the initial value of xi(t) at t ∈ [q, 1]and x(0) = (x10, x20, · · · , xn0)T , Ii(t) is the external input,fi(.), gi(.) and hi(.) are the nonlinear activation functions,and satisfy the following conditions:

fi(.), gi(.), hi(.) : R→ R,|fi(u)− fi(v)| ≤ Li|u− v|, |fi(u)| ≤ qi,|gi(u)− gi(v)| ≤ Mi|u− v|, |gi(u)| ≤ ri,|hi(u)− hi(v)| ≤ Ni|u− v|, |fi(u)| ≤ si,

(2)

where i = 1, 2, · · · , n, u, v ∈ R and Li,Mi, Ni, qi, ri and siare non-negative constants. By applying Brouwer fixed pointtheorem and constructing the delay differential inequality,Zhou and Zhang [31] obtained some delay-independent anddelay-dependent sufficient conditions to ensure the existence,uniqueness and global exponential stability of equilibrium of

IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_05

(Advance online publication: 26 August 2016)

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system (1). Moreover, the exponentially convergent rate isestimated.

In this paper, we will further investigate the exponentialstability of the following system

ẋi(t) = −dixi(t) +n∑

j=1

[aijfj(xj(t)) + bijgj(xj(q̃t))

+ cijhj(xj(q̃t))] + Ii,xi(s) = xi0, s ∈ [q̃t0, t0],

(3)which is a revised version of system (1). Here for simplifi-cation, we let q1 = q2 = q̃ in system (1), t0 is a constant.If t0 = 0, s ∈ [q̃t0, t0] is equal to s = t0 = 0. Differentfrom the work of Zhou and Zhang [31], we will obtain somesufficient conditions to ensure the pth exponential stabilityof system (3) by applying matrix measure and generalizedHalanay inequality. The results of this paper are completelynew and complement those of the previous studies in [31].The approach is new.

The organization of the rest of this paper is as follows.In Section 2, some preliminaries are presented. In Section3, some sufficient conditions are derived for the exponentialstability of (3) by matrix measure and generalized Halanayinequality. In Section 4, we present three examples to illus-trate the feasibility and effectiveness of our main theoreticalfindings in previous sections. A brief conclusion is drawn inSection 5.

II. PRELIMINARY RESULTS

First we give some notations. Let

x(t) = (x1(t), x2(t), · · · , xn(t))T ,D = diag{d1, d2, · · · , dn},

A = (aij)n×n, B = (bij)n×n, C = (cij)n×n,

f(x(t)) = (f1(x1(t)), f2(x1(t)), · · · , fn(x1(t))T ,g(x(t)) = (g1(x1(t)), g2(x1(t)), · · · , gn(x1(t))T ,h(x(t)) = (h1(x1(t)), h2(x1(t)), · · · , hn(x1(t))T ,

I = (I1, I2, · · · , In)T . Then system (3) can be rewritten asfollows:

ẋ(t) = −Dx(t)+ [Af(x(t))+Bg(x(q̃t))+Ch(x(q̃t))]+ I.(4)

Using the transformation

yi(t) = xi(et), i = 1, 2, · · · , n, (5)and letting y(t) = (y1(t), y2(t), · · · , yn(t))T , τ = − ln q̃ >0, we have

ẏ(t) = et{−Dy(t) + [Af(y(t)) + Bg(y(t− τ))+ Ch(y(t− τ))] + I}, (6)

with the initial condition

yi(s) = ϕi(s), t0 − τ ≤ s ≤ t0, i = 1, 2, · · · , n, (7)where ϕi(s) ∈ C([t0 − τ, t0],R) is a continuous function.In addition, we need the following definitions and lemmas.

Definition 2.1 An equilibrium point x∗ = (x∗1, x∗2, · · · , x∗n)Tof the system (8) is said to be pth (p = 1, 2,∞) globally

exponentially stable, if there exist two positive constantsM > 0 and λ > 0 such that ||x(t, t0, x0) − x∗||p ≤M ||x0 − x∗||pe−λt holds, where x0 = (x10, x20, · · · , xn0)Tis the initial condition of the system (8), x(t, t0, x0) is thesolution of system (8).

Definition 2.2 ([32]) For any real matrix A = (aij)n×n, itsmatrix measure is defined as

µp(A) = limε→0+

||E + εA||p − 1ε

,

where ||.||p denotes the matrix norm in Rn×n, E is theidentity matrix, p ∈ {1, 2,∞}.Let the matrix norm be as follows:

||A||1 = maxj

{n∑

i=1

|aij |}

, ||A||2 =√

λmax(AT A),

||A||∞ = maxI

{n∑

J=1

|aij |}

.

Then we get

µ1(A) = maxj

ajj +

n∑

i=1,i6=j|aij |

,

µ2(A) =12λmax(AT + A),

µ∞(A) = maxi

aii +

n∑

ji=1,j 6=i|aij |

.

Lemma 2.1 ([32]) For the definition of matrix measure, forany A,B ∈ Rn×n, p = 1, 2,∞, we have(1) −||A||p ≤ µp(A) ≤ ||A||p; (2) µp(αA) = αµp(A),∀α >0; (3) µp(A + B) ≤ µp(A) + µp(B).Lemma 2.2 ([33], Ageneralized Halanay,s inequality)Suppose

ẋ(t) ≤ γ(t)− α(t)x(t) + β(t) supt−τ≤σ≤t

x(σ)

holds for any t ≥ t0. Here τ ≥ 0, and γ(t), α(t), β(t)are continuous functions such that 0 ≤ γ(t) ≤ γ∗, α(t) ≥α0, 0 ≤ β(t) ≤ q̃α(t) for any t ≥ t0 with constantsγ∗ > 0, α0 > 0, 0 ≤ q < 1. Then we have

x(t) ≤ γ∗

(1− q̃)α0 + Ge−µ∗(t−t0)

holds for t ≥ t0. Here G = supt0−τ≤t≤t0 x(t) and µ∗ > 0is defined as

µ∗ = inft≥t0

{µ(t) : µ(t)− α(t) + β(t)eτµ(t) = 0}.

In Lemma 2.2, Letting γ(t) = 0, γ∗ → 0, then we can obtainthe following lemma.

Lemma 2.3 Suppose

ẋ(t) ≤ −α(t)x(t) + β(t) supt−τ≤σ≤t

x(σ)

holds for any t ≥ t0. Here τ ≥ 0, and α(t), β(t) arecontinuous functions such that α(t) ≥ α0, 0 ≤ β(t) ≤ qα(t)



______________________________________________________________________________________

for any t ≥ t0 with constants α0 > 0, 0 ≤ q < 1. Then wehave

x(t) ≤ Ge−µ∗(t−t0)

for t ≥ t0, where G = supt0−τ≤t≤t0 x(t) and µ∗ > 0 isdefined as

µ∗ = inft≥t0

{µ(t) : µ(t)− α(t) + β(t)eτµ(t) = 0}.

In order to obtain the main results, we make the followingassumptions.

(H1) For i = 1, 2, · · · , n, there exist positive constants αi, βiand γi such that |fi(u)−fi(v)| ≤ αi|u−v|, |gi(u)−gi(v)| ≤βi|u− v|, |hi(u)− hi(v)| ≤ γi|u− v| for all u, v ∈ R.Denote

α = max1≤i≤n

{αi}, β = max1≤i≤n

{βi}, γ = max1≤i≤n

{γi}.

III. MAIN RESULTS

In this section, we consider the global exponential stability ofsystem (4) by applying the matrix norm and matrix measure.

Theorem 3.1 Under the condition (H1), let Θ1 = −µp(−D)and Θ2 = α||A||p + β||B||p + γ||C||p. If Θ1 > Θ2 > 0,then the equilibrium point y∗ of system (4) is pth globallyexponentially stable.

Proof Assume that y∗ = (y∗1 , y∗2 , · · · , y∗n)T is the equilibriumpoint of system (6), then y∗ satisfies

−Dy∗ + [Af(y∗) + Bg(y∗) + Ch(y∗)] + I = 0.Let u(t) = y(t)− y∗, then

u̇(t) = et{−Du(t) + A[f(y(t))− f(y∗]+B[g(y(t− τ))− g(y∗)]+C[h(y(t− τ))− h(y∗)]}. (8)

Consider the following nonnegative function

V (t) = ||u(t)||p. (9)Calculating the derivative of V (t) along the trajectories of(9) leads to

D+V (t) = lim²→0+

||(u(t + ²)||p − ||u(t)||p²

= lim²→0+

||(u(t) + ²u̇(t) + o(²)||p − ||u(t)||p²

≤ lim²→0+

E + ²et(−D)||p − 1²

||u(t)||p+et{A[f(y(t))− f(y∗]+B[g(y(t− τ))− g(y∗)]+C[h(y(t− τ))− h(y∗)]}. (10)

In view of (H1), we get||f(y(t))− f(y∗)||p ≤ α||u(t)||p,||g(y(t− τ))− g(y∗)||p ≤ β||u(t− τ)||p,||h(y(t− τ))− h(y∗)||p ≤ γ||u(t− τ)||p.

(11)

It follows from (10) and (11) that

D+V (t) ≤ µp(−etD)||u(t)||p + et(α||A||p+β||B||p + γ||C||p)||u(t− τ)||p

≤ −Θ1et||u(t)||p+Θ2et sup

t−τ≤s≤t||u(s)||p. (12)

By Lemma 2.3, we have

V (t) ≤ supt0−τ≤s≤t0

V (s)e−µ∗(t−t0), (13)

where

µ∗ = inft≥t0

{µ(t) : µ(t)−Θ1et + Θ2et+τµ(t) = 0} > 0.

Then the zero solution of system (8) is pth globally expo-nentially stable, i.e., the equilibrium point y∗ of system (4)is pth globally exponentially stable. The proof of Theorem3.1 is complete.

IV. EXAMPLES

In this section, we give three examples to illustrate our mainresults derived in previous sections. Consider the followingthree cellular neural networks with proportional delays

Example 4.1 Consider the following cellular neural networkswith proportional delays

ẋ1(t) = −d1xi(t) +3∑

j=1

[a1jfj(xj(t)) + b1jgj(xj(q̃t))

+ c1jhj(xj(q̃t))] + I1,

ẋ2(t) = −d2x2(t) +2∑

j=1


+ c2jhj(xj(q̃t))] + I2,

ẋ3(t) = −d3x3(t) +3∑

j=1


+ c3jhj(xj(q̃t))] + I3,(14)

where

d1 0 00 d2 00 0 d3

=

0.3 0 00 0.1 00 0 0.5

,

a11 a12 a13a21 a22 a23a31 a32 a33

=

0.4 0.4 0.40.1 0.8 0.20.2 0.5 0.3

,

b11 b12 b13b21 b22 b23b31 b32 b33

=

0.5 0.6 0.10.8 0.3 0.50.5 0.4 0.2

,

c11 c12 c13c21 c22 c23c31 c32 c33

=

0.2 0.6 0.10.5 0.4 0.60.1 0.6 0.5

,

.

I1I2I3

=

0.20.50.7

, fi(x) = 12(|x + 1| − |x1|),

gi(x) = tanh(57x), hi(x) = tanh(

25x)(i = 1, 2, 3).

Set q̄ = 0.5 Then α = αi = 1, β = βi = 57 , γ = γi =25 .

It is easy to verify that Θ1 = −µp(−D) = 9.0703 and



______________________________________________________________________________________

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

1(t)

x2(t)

x3(t)

x1(t)

x2(t)

x3(t)

Fig. 1. Transient response of state variables x1(t), x2(t) and x3(t).

Θ2 = α||A||p + β||B||p + γ||C||p = 6.3423. It followsthat Θ1 > Θ2 > 0. Then all the conditions (H1)-(H2) ofTheorem 3.1 hold. Thus system (14) has a unique equilibrium(0.0921, 0.3532, 0.2036) which is globally exponentially sta-ble. The results are illustrated in Fig. 1.


ẋ1(t) = −d1xi(t) +3∑

j=1


+ c1jhj(xj(q̃t))] + I1,

ẋ2(t) = −d2x2(t) +3∑

j=1


+ c2jhj(xj(q̃t))] + I2,

ẋ3(t) = −d3x3(t) +3∑

j=1


+ c3jhj(xj(q̃t))] + I3,(15)

where

d1 0 00 d2 00 0 d3

=

0.8 0 00 0.4 00 0 0.7

,

a11 a12 a13a21 a22 a23a31 a32 a33

=

0.2 0.5 0.70.2 0.3 0.50.7 0.4 0.8

,

b11 b12 b13b21 b22 b23b31 b32 b33

=

0.9 0.3 0.70.3 0.8 0.70.8 0.1 0.5

,

c11 c12 c13c21 c22 c23c31 c32 c33

=

0.1 0.9 0.40.7 0.3 0.40.8 0.9 0.9

,

I1I2I3

=

0.60.90.2

, fi(x) = 12(|x + 1| − |x1|),


59x)(i = 1, 2, 3).



0 10 20 30 40 50 600.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

x

1(t)

x2(t)

x3(t)

x3(t)

x2(t)

x1(t)




ẋ1(t) = −d1xi(t) +3∑

j=1


+ c1jhj(xj(q̃t))] + I1,

ẋ2(t) = −d2x2(t) +3∑

j=1


+ c2jhj(xj(q̃t))] + I2,

ẋ3(t) = −d3x3(t) +3∑

j=1


+ c3jhj(xj(q̃t))] + I3,(16)

where

d1 0 00 d2 00 0 d3

=

0.5 0 00 0.3 00 0 0.5

,

a11 a12 a13a21 a22 a23a31 a32 a33

=

0.5 0.9 0.30.5 0.8 0.40.7 0.4 0.7

,

b11 b12 b13b21 b22 b23b31 b32 b33

=

0.2 0.5 0.50.8 0.5 0.40.8 0.7 0.6

,

c11 c12 c13c21 c22 c23c31 c32 c33

=

0.1 0.7 0.70.3 0.5 0.80.7 0.5 0.6

,

I1I2I3

=

0.60.30.8

, fi(x) = 12(|x + 1| − |x1|),


34x)(i = 1, 2, 3).





______________________________________________________________________________________

0 10 20 30 40 50 60−0.5

0

0.5

1

1.5

2

x

1(t)

x2(t)

x3(t)x3(t)

x1(t)

x2(t)



V. CONCLUSIONS

In this paper, we have investigated the global exponentialstability of cellular neural networks with proportional delays.Applying matrix measure and generalized Halanay inequal-ity, a series of new sufficient conditions to guarantee thepth exponential stability of cellular neural networks withproportional delays are established. The obtained conditionsare easily to check in practice. Finally, three examples areincluded to illustrative the feasibility and effectiveness. Tothe best of our knowledge, there are no results on theanti-periodic solution and synchronization for cellular neuralnetworks with proportional delays, which might be our futureresearch topic.

ACKNOWLEDGMENT

The authors would like to thank the anonymous refereesfor their helpful comments and valuable suggestions, whichled to the improvement of the manuscript.

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