Systems & Control Letters 58 (2009) 461–467

Contents lists available at ScienceDirect

Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Observations on the stability properties of cooperative systemsOliver Mason ∗, Mark VerwoerdHamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland

a r t i c l e i n f o

Article history:Received 8 September 2008Received in revised form7 February 2009Accepted 9 February 2009Available online 12 March 2009

Keywords:Positive systemsCooperative systemsGlobal asymptotic stabilityDelayD-stability

a b s t r a c t

We extend two fundamental properties of positive linear time-invariant (LTI) systems to homogeneouscooperative systems. Specifically, we demonstrate that such systems are D-stable, meaning that globalasymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneouscooperative system is globally asymptotically stable (GAS) for all non-negative delays if and only if thesystem is GAS for zero delay.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Positive dynamical systems, in which the state variablesare constrained to remain non-negative, arise in numerousapplication areas. In fact, any situation in which the variablesof interest take only non-negative values gives rise to a positivedynamical system. Examples of this type can be found inareas such as Ecology (population sizes), Biology (gene/proteinconcentrations), Economics (commodity prices) and ChemicalEngineering (chemical concentrations).Essentially a positive dynamical system is one for which

non-negative initial conditions always give rise to non-negativetrajectories. Given their practical importance, it is not surprisingthat a considerable deal of attention has been paid to thestudy of positive systems and to the elucidation of their basicproperties. At the time of writing, there is a well-developedtheory of positive linear time-invariant (LTI) systems, with rootsin the Perron–Frobenius theory of non-negative matrices [1,2].Particular attention has been paid to questions relating to positivereachability and observability [3,4], and the existence of positiverealizations of transfer functions [5]. As with any system class,issues pertaining to stability and the existence and location ofequilibria are of fundamental importance [6]. The work describedhere focuses on this aspect of the theory of positive systems.In the linear time-invariant case, it is well known [1] that if the

origin is a globally asymptotically stable (GAS) equilibrium of the

∗ Corresponding author. Tel.: +353 1 708 6463; fax: +353 1 708 6269.E-mail address: oliver.mason@nuim.ie (O. Mason).

0167-6911/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2009.02.009

positive LTI system x(t) = Ax(t), then it is also aGAS equilibriumofx(t) = DAx(t) for any diagonal matrix Dwith positive elements onthe diagonal. This property is commonly referred to as D-stability.More recently, it was shown in [7] that for a positive linear delayedsystem x(t) = Ax(t) + Bx(t − τ), the origin is a GAS equilibriumfor all τ ≥ 0 if and only if it is a GAS equilibrium of the systemwith zero delay, x(t) = (A + B)x(t). The two main results ofthe current paper will show that these two properties of positiveLTI systems naturally extend to an important class of nonlinearpositive systems.Specifically, we shall show that these results extend to the

class of homogeneous cooperative systems. Cooperative systemshave been studied extensively, particularly in relation to biologicalapplications [8,9]. A key property of the systems consideredhere, which also holds for positive LTI systems, is that they aremonotone [8,10]. Essentially this means that if we consider twoinitial vectors x0 and y0 where x0 is less than y0 componentwise,then the trajectory starting from x0 remains less than that startingfrom y0 (componentwise) for all subsequent times. Recently,motivated by applications in Cell Biology, the basic theory ofmonotone dynamical systems has been extended to considerinterconnections of such systems and a control theory ofmonotonesystems has been developed in the papers [11,12]. While theresult for delayed positive linear systems in [7] was derived usingLyapunov–Krasovskii techniques, the methods adopted here arebased on the fundamental monotonicity properties of cooperativesystems. As such, in addition to extending the result in [7], weprovide an alternative view on it.Other recent work on nonlinear positive systems has been

presented in [13]. In this paper, the stability and dissipativity

462 O. Mason, M. Verwoerd / Systems & Control Letters 58 (2009) 461–467

properties of nonlinear, not necessarily cooperative, positive sys-tems have been investigated. Further, motivated by applications inanaesthesiology, adaptive control methods for nonlinear positivesystems have been proposed and analysed in [14].The layout of the paper is as follows. In the Section 2,

we introduce notation and some mathematical background. InSection 3weprove that homogeneous cooperative systemspossessthe D-stability property. In Section 4 we consider systems subjectto delay and present a nonlinear extension of the result of [7].Finally, in Section 5 we give some concluding remarks.

2. Notation and background

Throughout thepaper,R andRn denote the field of real numbersand the vector space of all n-tuples of real numbers respectively.Rn×n denotes the space of n × n matrices with real entries. Forx ∈ Rn and i = 1, . . . , n, xi denotes the ith coordinate of x.Similarly, for A ∈ Rn×n, aij denotes the (i, j)th entry of A.In the interest of brevity, we shall slightly abuse notation and

refer to a system as being GASwhen the origin is a GAS equilibriumof the system. Also, as we are dealing with positive systemsthroughout, when we refer to a system as GAS, it is with respectto initial conditions in Rn

+, where Rn

+is the set of all vectors in Rn

with non-negative entries:Rn+:= {x ∈ Rn : xi ≥ 0, 1 ≤ i ≤ n}.

For vectors x, y ∈ Rn, we write: x ≥ y if xi ≥ yi for 1 ≤ i ≤ n;x > y if x ≥ y and x 6= y; x� y if xi > yi, 1 ≤ i ≤ n.A matrix A ∈ Rn×n is said to be non-negative if aij ≥ 0 for

1 ≤ i, j ≤ n. Similarly, a vector field g : Rn → Rn is non-negativeif g(x) ≥ 0 for all x ∈ Rn

+.

For a real interval [a, b], C([a, b],Rn+) denotes the space of all

real-valued continuous functions on [a, b] taking values in Rn+. For

functions f , g ∈ C([a, b],Rn+) we write f ≥ g if f (s) ≥ g(s), ∀s ∈

[a, b].Throughout the paper, for a vector x ∈ Rn, ‖x‖denotes the usual

Euclidean norm of x. For amatrix A ∈ Rn×n, ‖A‖ denotes thematrixnorm induced by the Euclidean norm. Finally, for d = (d1, . . . , dn)Tin Rn, diag(d1, . . . , dn) denotes the diagonal matrix in Rn×n inwhich the ith entry on its main diagonal is di.

2.1. Positive linear systems

In this subsection, we recall some basic facts concerningpositive LTI systems.

Definition 2.1. The LTI system

x(t) = Ax(t) (1)

is positive if x0 ≥ 0 ⇒ x(t, x0) ≥ 0 for all t ≥ 0, where x(·, x0)denotes the unique solution of (1) satisfying x(0, x0) = x0.It is well known and easily verified that an LTI system is positive ifand only if its systemmatrix A satisfies aij ≥ 0 for i 6= j. Matrices ofthis form are said to beMetzler.1 Note that it follows from linearitythat if x0 ≤ y0, then x(t, x0) ≤ x(t, y0) for all t ≥ 0. Thus, positiveLTI systems are automatically monotone.For convenience, we collect some standard facts concerning the

stability of positive LTI systems in the following result.

Theorem 2.1. Consider the positive LTI system (1). The followingstatements are equivalent:(i) The system (1) is globally asymptotically stable;(ii) There is some vector v � 0 such that Av � 0;(iii) x(t) = DAx(t) is globally asymptotically stable for all diagonal

matrices D = diag(d1, . . . , dn) with di > 0 for 1 ≤ i ≤ n.

1 In the mathematics literature, it is more common to refer to such matrices asessentially non-negative.

The property described by point (iii) above is usually referred to asD-stability.In the recent papers [7,15] the stability properties of delayed

positive linear systems were studied. Formally, consider thesystemx(t) = Ax(t)+ Bx(t − τ), τ ≥ 0. (2)Recall that for delay systems of this form [16], for any initialconditions given by a function φ ∈ C([−τ , 0],Rn

+) there exists

a unique solution x(t, φ) defined for t ∈ [−τ ,∞), satisfying (2)and x(s, φ) = φ(s) for s ∈ [−τ , 0]. Typically, the history segmentxt : [−τ , 0] → Rn given by xt(s) = x(t − s) for −τ ≤ s ≤ 0 isreferred to as the state of the system at time t .As with LTI systems, the system (2) is positive if φ ≥ 0 implies

x(t, φ) ≥ 0 for all t ≥ 0. It has been shown in [7] that (2) is positiveif and only if A isMetzler and B is non-negative. As in the undelayedcase, it follows from linearity that the flows generated by a positivedelayed linear system are monotone meaning that φ ≤ ψ impliesx(t, φ) ≤ x(t, ψ) for all t ≥ 0.The following result from [7] shows that (2) is globally

asymptotically stable for any τ ≥ 0 if and only if the undelayedsystem with τ = 0 is globally asymptotically stable.

Theorem 2.2. The time-delayed positive linear system (2) is globallyasymptotically stable for all τ ≥ 0 if and only if

x(t) = (A+ B)x(t) (3)

is globally asymptotically stable.

Comment. Theorem 2.2 is a slight rewording of Theorem 3.1of [7]. A closely related result concerning the exponential stabilityof delay systems subsequently appeared as Theorem 4.1 of [15].The proof in [7] relies on the theory of Lyapunov–Krasovskiifunctionals, while the proof for exponential stability in [15] isbased on analysing the characteristic function of the delayedsystem.

2.2. Homogeneous cooperative systems

Cooperative systems are of particular importance for numerousapplications in Economics, Biology and Ecology and have attracteda considerable deal of attention in the past [8]. The definition of acooperative vector field is as follows.

Definition 2.2. A continuous vector field f : Rn → Rn, which is C1

onRn\{0} is said to be cooperative if the Jacobian ∂ f∂x (a) is aMetzler

matrix for all a ∈ Rn+\{0}.

The results presented later in the paper are concerned withcooperative systems whose vector fields are homogeneous in thesense of the following definition.

Definition 2.3. f : Rn → Rn is said to be homogeneous if for allx ∈ Rn and all real λ > 0, f (λx) = λf (x).Let the vector field f : Rn → Rn be continuous on Rn and C1on Rn\{0}. If f is homogeneous then it follows easily that for anya ∈ Rn\{0} and any λ > 0,

∂ f∂x(λa) =

∂ f∂x(a). (4)

This immediately allows us to conclude the following result, whoseproof we include in the interest of completeness.

Lemma 2.1. Let f : Rn → Rn be continuous on Rn, C1 on Rn\{0}and homogeneous. Then there exists K > 0 such that ‖f (x)− f (y)‖ ≤K‖x− y‖ for all x, y ∈ Rn.Proof. As f is C1, it follows that there is some K1 > 0 such that∥∥∥∥∂ f∂x (a)

∥∥∥∥ ≤ K1

O. Mason, M. Verwoerd / Systems & Control Letters 58 (2009) 461–467 463

for all awith ‖a‖ = 1. But (4) then implies that this inequalitymusthold for all a 6= 0. Now pick any x 6= 0 in Rn. It follows from themean value theorem that for any y ∈ Rn\{tx : t ≤ 0},

‖f (x)− f (y)‖ ≤ K1‖x− y‖. (5)

It now follows from the continuity of f that this inequality musthold for all y ∈ Rn. As xwas an arbitrary non-zero vector, it followsthat (5) holds for all x ∈ Rn\{0}, y ∈ Rn. It now follows again bycontinuity that the inequalitymust hold for all x, y ∈ Rn. The resultnow follows with K = K1. �

If f : Rn → Rn is homogeneous, the previous lemmaimmediately implies the existence and uniqueness of solutions tothe system x(t) = f (x(t)) for any initial conditions in Rn.Wenext collect somewell-established facts concerning dynam-

ical systems defined by cooperative, homogeneous vector fields.Before stating the result, note that Lemma 5.1 of [10] establishedthe existence of solutions to systems with cooperative, homoge-neous right-hand sides for all t ≥ 0.

Theorem 2.3. Let f : Rn → Rn be a vector field that is continuouson Rn and C1 on Rn\{0} and suppose that f is homogeneous andcooperative. Consider the associated dynamical system

x(t) = f (x(t)) (6)

and for x0 ∈ Rn+, let x(·, x0) denote the solution of (6) satisfying

x(0, x0) = x0. Then:(i) For any real number λ > 0 and any x0 ∈ Rn

+, x(t, λx0) =

λx(t, x0) for any t ≥ 0;(ii) For any x0, x1 ∈ Rn

+, if x0 ≤ x1, it follows that x(t, x0) ≤ x(t, x1)

for any t ≥ 0.Proof. Under the hypotheses of the theorem, for any x0 ∈ Rn

+,

there exists a unique solution of (6) through x0. This togetherwith the homogeneity of f implies (i). The statement in (ii) wasproven in Proposition 4.3 of [10] and also follows from the moregeneral results stated as Proposition 3.1.1 in [8] or Theorem 16.2of [17]. �

Finally for this section, we state a result for delayed systemsthat corresponds to point (ii) of Theorem 2.3 above. For detailsconsult [8]. First of all, we recall the definition of an order-preserving vector field.

Definition 2.4. g : Rn → Rn is order-preserving on Rn+if g(x) ≥

g(y) for any x, y ∈ Rn+such that x ≥ y.

As we shall only be interested in vector fields that are order-preserving on Rn

+, we shall usually refer to such vector fields as

simply order-preserving.

Theorem 2.4. Let f , g : Rn → Rn be continuous on Rn and C1on Rn\{0}. Further assume that f is cooperative and homogeneouswhile g is order-preserving and homogeneous. Let x(·, φ) denote thesolution of the delayed system

x(t) = f (x(t))+ g(x(t − τ)) (7)

corresponding to the initial condition φ ∈ C([−τ , 0],Rn+). Then for

any φ,ψ ∈ C([−τ , 0],Rn+) with φ ≤ ψ , we have that x(t, φ) ≤

x(t, ψ) for all t ≥ 0 for which both solutions are defined.

Comment. As in the case of undelayed systems, the existence anduniqueness of solutions to (7) is implied by Lemma2.1 (see Chapter2 of [16]).

2.3. A nonlinear Perron–Frobenius theorem

Many of the stability properties of positive linear systemsare natural consequences of the Perron–Frobenius theorem fornon-negative matrices. Numerous authors have considered theproblem of extending the Perron–Frobenius Theorem to nonlinearpositive systems. The appendix in [18] is an excellent early

reference on this topic (in finite dimensions), while further andmore general resultswere subsequently reported in [19,20,10].Weshall only require a particular case of a recent result presentedin [10] for irreducible, homogeneous cooperative systems.

Definition 2.5. f : Rn → Rn is said to be irreducible on Rn+if:

(i) ∂ f∂x (a) is an irreduciblematrix [21] for all a in the interior ofR

n+;

(ii) for non-zero a in the boundary ofRn+, either ∂ f

∂x (a) is irreducibleor ai = 0 implies fi(a) > 0.

The paper [10] contains a variety of interesting technicalresults concerning the asymptotic properties of homogeneouscooperative systems but, for our purposes, the facts collected inthe following theorem will prove sufficient.

Theorem 2.5. Let f : Rn → Rn be continuous on Rn and C1on Rn\{0}. Further, assume that f is cooperative, homogeneous andirreducible. Then there exist a unique vector v � 0 and a real numberγv such that f (v) = γvv. Moreover the system x(t) = f (x(t)) isglobally asymptotically stable if and only if γv < 0.

3. Stability and D-stability for the homogeneous cooperativesystems

In this section, we shall extend Theorem 2.1 to homogeneouscooperative systems. Throughout this section, unless statedotherwise, all vector fields are continuous on the whole of Rn andC1 on Rn\{0}. In proving the main result of this section, we shallneed the following technical lemma.

Lemma 3.1. Let f : Rn → Rn be cooperative and homogeneous andsuppose the associated system

x(t) = f (x(t)) (8)

is globally asymptotically stable. Then there exists an irreduciblecooperative, homogeneous vector field f1 : Rn → Rn such that:(i) f1(x) ≥ f (x) for all x ≥ 0;(ii) x(t) = f1(x(t)) is globally asymptotically stable.

Proof. Choose any homogeneous, irreducible, order-preservingvector field g : Rn → Rn. Note that for all ε > 0, f + εgis cooperative, homogeneous and irreducible. We claim that forε > 0 sufficiently small, the system x(t) = (f + εg)(x(t)) willbe globally asymptotically stable (GAS). Suppose this is not true.Then we can choose a sequence of positive real numbers εn suchthat εn → 0 as n → ∞ for which f + εng is not GAS for any n.It follows from Theorem 2.5 that there exist vectors vn � 0 with‖vn‖ = 1, and real numbers γn ≥ 0 such that for all n

(f + εng)(vn) = γnvn. (9)

As f and g are continuous and ‖vn‖ = 1 for all n, it follows thatthere is some positive constantM such that ‖(f + εng)(vn)‖ ≤ Mfor all n. Hence the sequence γn is bounded. As the sequences vnand γn are bounded, by passing to subsequences if necessary, wecan assume that vn → v for some v ≥ 0, ‖v‖ = 1 and that γn → γfor someγ ≥ 0. This togetherwith the continuity of f and g impliesthat f (v) = γ v. This implies that the solution x(·, v) of the systemx(t) = f (x(t)) with x(0) = v cannot tend to the equilibrium at0 as t → ∞, which contradicts our assumption that this originalsystem is GAS. This shows that there must be some ε1 > 0 forwhich x(t) = (f + ε1g)(x(t)) is GAS. It follows immediately thatf1 = f + ε1g satisfies (i) and (ii). �

From Theorem 2.1 we know that a positive LTI system x(t) =Ax(t) is globally asymptotically stable if and only if there is somev � 0 such that Av � 0. In the next result we show that this resultextends directly to homogeneous cooperative systems.

464 O. Mason, M. Verwoerd / Systems & Control Letters 58 (2009) 461–467

Theorem 3.1. Let f : Rn → Rn be cooperative and homogeneous.Then the system

x(t) = f (x(t)), (10)

is globally asymptotically stable if and only if there is some v � 0such that f (v)� 0.

Proof. Suppose that v � 0 satisfies f (v)� 0. We shall first showthat the trajectory x(t, v) starting from the initial state v tends tothe equilibrium at zero as t tends to infinity. As f (v)� 0 it followsthatdxdt(t, v)

∣∣∣∣t=0� 0

and hence there is some δ > 0 such that

x(t)� x(0) = v for all t ∈ (0, δ]. (11)

In particular, x(δ, v) � v and thus there exists a real number αwith 0 < α < 1 such that x(δ, v) ≤ αv.Now as f is cooperative, it follows from Theorem 2.3 that

x(2δ, v) = x(δ, x(δ, v))≤ x(δ, αv).

Further, as f is homogeneous, this in turn implies that

x(2δ, v) ≤ x(δ, αv)= αx(δ, v) ≤ α2v.

Moreover, we can also conclude from the cooperativity of f and(11) that x(t) � αv for all t ∈ (δ, 2δ). Iterating, we see that forp = 2, 3, . . .

x(pδ) ≤ αpvx(t)� αp−1v for t ∈ ((p− 1)δ, pδ)

and hence x(t, v)→ 0 as t →∞ as claimed.Now, let x0 ∈ Rn

+be given. As v � 0, we can always choose

some positive real number K such that x0 � Kv. As f is assumedto be cooperative inRn

+, it follows fromTheorem2.3 that x(t, x0) ≤

x(t, Kv) = Kx(t, v) for all t ≥ 0. But it now follows from the aboveargument that x(t, x0) → 0 as t → ∞. Hence, the system (10) isglobally asymptotically stable as claimed.Conversely, suppose that (10) is globally asymptotically stable.

By Lemma 3.1, we can choose some irreducible, homogeneous,cooperative mapping f1 : Rn → Rn, satisfying

f1(x) ≥ f (x) for all x ∈ Rn+,

such that the system x(t) = f1(x(t)) is GAS. Theorem 2.5 thenimplies that there is some vector v � 0 such that f1(v) � 0. Butf (v) ≤ f1(v) and hence f (v)� 0 aswell. This completes the proof.

�

Comment. In [22], a result similar to Theorem 3.1 for discrete-time systems has been presented. The authors of this manuscripthave shown, under the assumption that f : Rn → Rn is irreducibleand order-preserving, that x(k + 1) = f (x(k)) is GAS if and onlyif there is no non-zero w ∈ Rn

+satisfying f (w) ≥ w. Note that

Theorem 3.1 combined with Proposition 3.1 establishes that in thecontinuous-time case, GAS is equivalent to there being no non-zerow ∈ Rn

+satisfying f (w) ≥ 0. In this context, it is also worth noting

the earlier, biologically motivated work of [23] on the asymptoticbehaviour of homogeneous, order-preserving recursions.

Example 3.1. Consider the system x(t) = f (x(t)) defined on R2where

f (x1, x2) :=

2x2 − x1 −√x21 + x222x1 − 2x2 −

√x21 + x

22

.

Fig. 1. Phase portrait of f (x1, x2) := (2x2−x1−√x21 + x

22, 2x1−2x2−

√x21 + x

22)T .

A phase portrait of this system is given in Fig. 1. Two trajectories ofthe system are also illustrated, starting from the respective initialconditions x(1) := (x(1)1 , x

(1)2 ) = (0.2, 0.8) (solid line) and x(2) :=

(x(2)1 , x(2)2 ) = (0.8, 0.2) (dashed line). Note that f (x(i)) 6� 0 for

i = 1, 2. However, it is clear from the figure that there exist (x1, x2)such that f (x1, x2) � 0 (take for instance x1 = x2 = 1

2 ). Hence byTheorem 3.1, the system x = f (x) is GAS.

Comment. For positive LTI systems it is well known that if x(t) =Ax(t) is globally asymptotically stable, then x(t) = Bx(t) willalso be GAS for all Metzler matrices B with B ≤ A. The aboveresult allows us to immediately draw the corresponding, known,conclusion for homogeneous cooperative systems.

Corollary 3.1. Let f : Rn → Rn be a cooperative, homogeneousvector field and suppose that there is some v � 0 with f (v) � 0.Let g : Rn → Rn be a vector field such that the system

x(t) = g(x(t))

leaves Rn+invariant and such that g(x) ≤ f (x) for all x ∈ Rn

+. Then

the system

x(t) = g(x(t))

is globally asymptotically stable.

Proof. It follows from Theorem 3.1 that x(t) = f (x(t)) is GAS. Theresult now follows immediately from either Theorem 16.2 of [17]or Theorem 5.1.1 of [8]. �

For a positive LTI system x(t) = Ax(t), it is well known thatglobal asymptotic stability implies that the diagonal entries of Aare all negative. The following simple corollary can be seen as anextension of this fact to homogeneous systems.

Corollary 3.2. Let f : Rn 7→ Rn be cooperative and homogeneousand suppose the system x = f (x) is globally asymptotically stable.Then there exists v such that ∂ fi(ξ)

∂ξi|ξ=v < 0 for all i.

Proof. Suppose the system is GAS; then there exists v � 0 suchthat f (v) � 0. Euler’s formula for homogeneous functions (seeSection 3.1 of [10] for the general form of this classical result)implies that(∑j6=i

∂ fi(ξ)∂ξj

∣∣∣∣ξ=v

vj

)+∂ fi(ξ)∂ξi

∣∣∣∣ξ=v

vi = fi(v) (12)

O. Mason, M. Verwoerd / Systems & Control Letters 58 (2009) 461–467 465

for i = 1, . . . , n. As v � 0 the first term on the left-hand side of (12) is non-negative by assumption of cooperativity.Furthermore, as f (v) � 0, it follows that ∂ fi(ξ)

∂ξi|ξ=v < 0 for

i = 1, . . . , n. This completes the proof. �

Comment. Theorem 3.1 provides a test for the global asymptoticstability of a homogeneous cooperative system; If we candemonstrate the existence of a vector v � 0 with f (v) � 0, thenthe associated system is GAS. Conversely if no such vector exists,then the system is not GAS. However, it is typically much harderto show conclusively that no vector v � 0 satisfying f (v) � 0exists. The following result, which can be thought of as a nonlineartheorem of the alternative, provides a way of demonstrating that asystem is definitely not GAS.

Proposition 3.1. Let f : Rn → Rn be homogeneous and cooperative.Then exactly one of the following statements is true.(i) There is some v � 0 with f (v)� 0;(ii) There is some non-zerow ≥ 0 with f (w) ≥ 0.Proof. We first show that at most one of these statements can betrue. By way of contradiction, assume that there is some v � 0such that f (v) � 0 and some non-zero w ≥ 0 with f (w) ≥ 0.Now let λ = max1≤i≤n(wi/vi). As w 6= 0, λ > 0. Moreover, if wedefine v′ = λv, it follows that f (v′)� 0, v′ ≥ w and that there issome index p for whichwp = v′p. Then for this p,

fp(v′)− fp(w) =∫ 1

0

dds(fp(sv′ + (1− s)w))ds

=

∫ 1

0

(n∑j=1

∂ fp∂xj(sv′ + (1− s)w)(v′j − wj)

)ds

≥ 0,

where the final inequality follows from the cooperativity of f , v′ ≥w andwp = v′p. But the above implies that

0 > fp(v′) ≥ fp(w) ≥ 0,

which is a contradiction.The final step in the proof is to show that at least one of the

statements (i), (ii) must be true. Suppose that (i) is not true. Thenby Theorem 3.1, x(t) = f (x(t)) is not GAS. Let g : Rn → Rn bean irreducible, homogeneous, order-preserving vector field. Then,as (f + εg)(x) ≥ f (x) for all ε > 0 and all x ≥ 0, it followsimmediately from Corollary 3.1 that the system x(t) = (f +εg)(x(t)) is not GAS. Now applying Theorem 2.5 we can concludethat for every ε > 0, there exist some non-zero wε ≥ 0 with(f +εg)(wε) ≥ 0. Now, by suitably adapting the limiting argumentof Lemma 3.1, we can conclude that there must exist some non-zerow ≥ 0 with f (w) ≥ 0. �

Comment. The previous result provides ameans of demonstratingthat a homogeneous cooperative system is not GAS by showing theexistence of a non-zerow ≥ 0 with f (w) ≥ 0.The so-called D-stability property of positive LTI systems

asserts that the global asymptotic stability of such systems ispreserved under positive diagonal scaling. Theorem 3.1 allows usto immediately show that this property extends to homogeneouscooperative systems.

Theorem 3.2. Let f : Rn → Rn be cooperative and homogeneous.The system

x(t) = f (x(t)) (13)

is globally asymptotically stable if and only if

x(t) = Df (x(t)) (14)

is globally asymptotically stable for any diagonal matrix D =

diag(d1, . . . , dn) with di > 0 for 1 ≤ i ≤ n.

Proof. Suppose that (13) is globally asymptotically stable and letthe diagonal matrix D = diag(d1, . . . , dn) with di > 0 for 1 ≤ i ≤n be given. Then by Theorem3.1 it follows that there is some v � 0such that f (v) � 0. But it is immediate that Df (v) � 0 and asDf : Rn → Rn is also homogeneous and cooperative it follows fromTheorem 3.1 that the system (14) is also globally asymptoticallystable. The converse direction is immediate. �

4. Stability of delayed cooperative systems

In this section, unless explicitly stated otherwise, all vectorfields are assumed to be continuous on Rn and C1 on Rn\{0}.The next result is a direct extension of Theorem 2.2 to

homogeneous systems. It also provides an alternative view onthe result of [7] as our argument is based on the fundamentalmonotonicity property of the trajectories of the system in contrastto the techniques used in that paper.

Theorem 4.1. Let f : Rn → Rn and g : Rn → Rn be homogeneousand assume that f is cooperative and g is order-preserving. Then thetime-delay system

x(t) = f (x(t))+ g(x(t − τ)) (15)

is globally asymptotically stable for all τ ≥ 0 if and only if theundelayed system

x(t) = f (x(t))+ g(x(t)) (16)

is globally asymptotically stable.

Proof. If the system (15) is globally asymptotically stable (GAS) forall τ ≥ 0 then obviously it is GAS for τ = 0.Conversely, suppose that (16) is GAS. Let τ ≥ 0 be given

and for any φ ∈ C([−τ , 0],Rn+), let x(·, φ) denote the solution

of (15) corresponding to the initial condition φ. It follows fromTheorem 3.1 that there is some vector v � 0 such that (f +g)(v)� 0. Let v denote the element of C([−τ , 0],Rn

+) given by

v(s) = v for − τ ≤ s ≤ 0.

The first step in the proof is to show that the solution x(t, v) of (15)corresponding to the initial condition v converges asymptoticallyto the equilibrium point 0 as t →∞.Now asddt(x(t, v))

∣∣∣∣t=0� 0,

as in the proof of Theorem 3.1, it follows that there is some δ > 0such that x(t, v)� v for all t ∈ (0, δ]. Further, if we write xδ,v forthe element in C([−τ , 0],Rn

+) given by

xδ,v(s) = x(δ − s, v) for − τ ≤ s ≤ 0,

we can conclude that xδ,v ≤ v.Now as f is cooperative and g is order-preserving, it follows

from Theorem 2.4 and the above observations that for all s ∈ (0, δ]

x(s+ δ, v) = x(s, xδ,v) ≤ x(s, v)� v. (17)

Repeating this process, we can conclude that for k = 2, 3, . . . andfor all s ∈ (0, δ],

x(s+ kδ, v)� v. (18)

If we choose k large enough to ensure that t1 = kδ > τ , then itfollows that x(s, v) � v for s ∈ [t1 − τ , t1]. Hence, as [t1 − τ , t1]is compact, there is some α with 0 < α < 1 such that

x(s, v) ≤ αv (19)

for all s ∈ [t1 − τ , t1]. In other words, xt1,v ≤ αv. Moreover,x(s, v) ≤ v for all s ∈ [0, t1].

466 O. Mason, M. Verwoerd / Systems & Control Letters 58 (2009) 461–467

(a) τ = 1. (b) τ = 2.

Fig. 2. Trajectories of the system (21) for different state histories and different values of the delay parameter τ .

It now follows from Theorem 2.4 and the homogeneity of f andg that for s ∈ [0, t1],

x(s+ t1, v) = x(s, xt1,v) ≤ x(s, αv) = αx(s, v).

Hence, x2t1,v ≤ αxt1,v ≤ α2v. Iterating, we get that for p =2, 3, 4, . . . .

xpt1,v ≤ αpv, (20)

and hence as 0 < α < 1, it follows that x(t, v)→ 0 as t →∞.To complete the proof, supposewe are given an initial condition

φ ∈ C([−τ , 0],Rn+). As φ is continuous and hence bounded on

[−τ , 0], by choosing M sufficiently large, we can ensure that φ ≤Mv. As f is cooperative and g is order-preserving, it follows fromTheorem 2.4 that x(t, φ) ≤ x(t,Mv). Moreover as f and g arehomogeneous, x(t,Mv) = Mx(t, v). It now follows immediatelyfrom the argument in the previous paragraph that x(t, φ)→ 0 ast →∞. �

Comment. While the above result has been proven for the single-delay case, the argument can readily be adapted to systems withmultiple delays of the form x(t) = f (x(t))+ g1(x(t − τ1))+ · · · +gp(x(t − τp)). As essentially the same argument goes through, wedo not provide full details here.

Note that the argument given above also establishes that thesystem (15) is GAS if and only if there is some vector v � 0 suchthat (f + g)(v) � 0. In analogy with the undelayed case, thisallows us to conclude the following using Theorem 5.1.1 of [8].

Corollary 4.1. Let f : Rn → Rn and g : Rn → Rn behomogeneous vector fields and suppose that f is cooperative and gis order-preserving. Further assume that there is some v � 0 suchthat (f + g)(v)� 0. Let f1 : Rn → Rn and g1 : Rn → Rn be vectorfields such that Rn

+is invariant under the dynamics

x(t) = f1(x(t))+ g1(x(t − τ))

and (f1 + g1)(x) ≤ (f + g)(x) for all x ∈ Rn+. Then the system

x(t) = f1(x(t))+ g1(x(t − τ))

is globally asymptotically stable for all τ ≥ 0.

Example 4.1. Consider the delayed system:

x(t) = f (x(t))+ g(x(t − τ)), (21)

where τ ≥ 0 is a delay parameter,

f (x1, x2) :=(−3 62 −2

)(x1x2

)−

√x21 + x

22

(31

), (22)

and

g(x1, x2) :=

x1x2√x21 + x

22

x1x2√2x21 + 3x

22

. (23)

It is straightforward to verify that this system satisfies theconditions of Theorem 4.1. Moreover, (f + g)(1, 1) � 0. Hence,we can conclude that the origin is a GAS equilibrium of this systemfor any τ ≥ 0.Several sample trajectories of the system (21) are shown in

Fig. 2.For i = 1, . . . , 6, let the initial conditions x(a,i) and x(b,i) be

defined as follows:

x(a,i)(s) :=

1.1 cos( π14i)+ 0.1 cos

(2π(t +32

)−π

14i)

1.1 sin( π14i)− 0.1 sin

(2π(t +32

)−π

14i) ,

s ∈ [−1, 0]; (24)

x(b,i)(s) :=

cos( π14i)+ 0.1 s (s+ 2)

sin( π14i)− 0.1 s (s+ 2)2

, s ∈ [−2, 0]. (25)

The panel on the left-hand side in Fig. 2 shows 6 trajectoriesof the system (21) with τ = 1, corresponding to the initialconditions given by (24). The panel on the right of the figure shows6 trajectories of the system (21) with τ = 2, corresponding to theinitial conditions given by (25). It can be seen from the figure thatall trajectories converge to the origin as expected.

5. Conclusions

In this paper we have presented a number of results thatextend fundamental properties of positive linear time-invariant(LTI) systems to a significant class of nonlinear positive systems.Specifically, we have shown that for homogeneous, cooperativesystems, the D-stability property of positive LTI systems holds. In

O. Mason, M. Verwoerd / Systems & Control Letters 58 (2009) 461–467 467

addition, we have demonstrated that a homogeneous cooperativesystem subject to delay is globally asymptotically stable (GAS)for every non-negative delay if and only if it GAS for zero delay.Examples have been presented to illustrate the main results.While the class of homogeneous and cooperative systems is in

itself quite restrictive, it seems probable that the results presentedhere can be extended to broader system classes. In particular, theauthors believe that the assumption of homogeneity may not benecessary for the two main results presented here. However, itseems likely that in order to relax this assumption, some additionalhypotheses, such as irreducibility may need to be added. This iscurrently the subject of ongoing research by the authors and moregeneral results should hopefully be derived in the future.

Acknowledgements

This work was partially supported by the Science FoundationIreland (SFI) grant 03/RP1/I382 and the Irish Higher EducationAuthority (HEA) PRTLI Network Mathematics grant. NeitherScience Foundation Ireland nor the Higher Education Authorityis responsible for any use of data appearing in this publication.The authors would like to thank the anonymous reviewers fortheir helpful comments which have considerably improved themanuscript.

References

[1] L. Farina, S. Rinaldi, Positive Linear Systems, Wiley Interscience Series, 2000.[2] A. Berman, R.J. Plemmon, Non-Negative Matrices in the MathematicalSciences, in: SIAM Classics in Applied Mathematics, 1994.

[3] R. Bru, S. Romero, E. Sanchez, Canonical forms for positive discrete-time linearcontrol systems, Linear Algebra and its Applications 310 (2000) 49–71.

[4] E. Valcher, Controllability and reachability criteria for discrete-time positivesystems, International Journal of Control 65 (3) (1996) 511–536.

[5] L. Benvenuti, L. Farina, A tutorial on the positive realization problem, IEEETransactions on Automatic Control 49 (5) (2004) 651–664.

[6] P. de Leenheer, D. Aeyels, Stability properties of equilibria of classes of co-operative systems, IEEE Transactions on Automatic Control 46 (12) (2001)1996–2001.

[7] W.M. Haddad, V. Chellaboina, Stability theory for nonnegative and compart-mental dynamical systems with time delay, Systems and Control Letters 51(2004) 355–361.

[8] H. Smith, Monotone Dynamical Systems, an Introduction to the Theory ofCompetitive and Cooperative Systems, in: AMS Mathematical Surveys andMonographs, 1995.

[9] M.W. Hirsch, Systems of differential equations that are competitive or cooper-ative. ii: Convergence almost everywhere, SIAM Journal ofMathematical Anal-ysis 16 (1985) 423–439.

[10] P. de Leenheer, D. Aeyels, Extension of the Perron–Frobenius theorem tohomogeneous systems, SIAM Journal of Control and Optimization 41 (2)(2002) 563–582.

[11] D. Angeli, E.D. Sontag, Multi-stability in monotone input/output systems,Systems and Control Letters 51 (3–4) (2004) 185–202.

[12] D. Angeli, E.D. Sontag, Monotone control systems, IEEE Transactions onAutomatic Control 48 (10) (2003) 1684–1698.

[13] W.M. Haddad, V. Chellaboina, Stability and dissipativity theory for nonnega-tive dynamical systems: A unified analysis framework for biological and physi-ological systems, Nonlinear Analysis: RealWorld Applications 6 (2005) 35–65.

[14] W.M. Haddad, T. Hayakawa, J.M. Bailey, Adaptive control for nonlinear com-partmental dynamical systems with applications to clinical pharmacology,Systems and Control Letters 55 (2006) 62–70.

[15] P.H.A. Ngoc, A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices, Applied Mathematics Letters 19 (8) (2006) 747–751.

[16] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional DifferentialEquations, Springer-Verlag, 1993.

[17] J. Szarski, Differential Inequalities, Monographie Matematyczne, Warsaw,1965.

[18] M. Morishima, Equilibrium, Stability and Growth — AMulti-sectorial Analysis,Oxford University Press, 1964.

[19] R.D. Nussbaum, Hilbert’s Projective Metric and Iterated Nonlinear Maps,Memoirs of the American Mathematical Society, 1988.

[20] S. Gaubert, J. Gunawardena, The Perron–Frobenius theorem for homogeneousmonotone functions, Transactions of the American Mathematical Society 356(12) (2004) 4931–4950.

[21] R. Horn, C. Johnson, Matrix Analysis, Cambridge University Press, 1985.[22] S. Dashkovskiy, B. Rüffer, F. Wirth, An ISS small-gain theorem for general

networks, Technical Report 2005-05, Zentrum für Technomathematik,Universiteit Bremen, 2005.

[23] R. Lui, Biological growth and spread modeled by systems of recursions i.mathematical theory, Mathematical Biosciences 93 (1989) 269–285.