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Systems & Control Letters 51 (2004) 355–361www.elsevier.com/locate/sysconle

Stability theory for nonnegative and compartmental dynamicalsystems with time delay�

Wassim M. Haddada ;∗, VijaySekhar Chellaboinab

aSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USAbMechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

Received 10 April 2002; received in revised form 19 August 2003; accepted 8 September 2003

Abstract

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerationsand involve the exchange of nonnegative quantities between subsystems or compartments. These models are widespread inbiological and physical sciences and play a key role in understanding these processes. A key physical limitation of such systemsis that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systemsshould account for material in transit between compartments. In this paper, we present necessary and su7cient conditions forstability of nonnegative and compartmental dynamical systems with time delay. Speci8cally, asymptotic stability conditionsfor linear and nonlinear nonnegative dynamical systems with time delay are established using linear Lyapunov–Krasovskiifunctionals.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Nonnegative systems; Compartmental systems; Stability theory; Time delay; Linear Lyapunov–Krasovskii functionals

1. Introduction

Nonnegative and compartmental models play a keyrole in understanding many processes in biologicaland medical sciences [1,6,9,12,13,20]. Such modelsare composed of homogeneous interconnected sub-systems (or compartments) which exchange variable

� This research was supported in part by the National Sci-ence Foundation under Grants ECS-9496249 and ECS-0133038and the Air Force O7ce of Scienti8c Research under GrantF49620-03-1-0178.

∗ Corresponding author. Tel.: +1-404-894-1078; fax: +1-404-894-2760.

E-mail addresses: wm.haddad@aerospace.gatech.edu(W.M. Haddad), Chellaboinav@missouri.edu (V. Chellaboina).

nonnegative quantities of material with conservationlaws describing transfer, accumulation, and outFowsbetween compartments and the environment. Therange of applications of nonnegative and compartmen-tal systems is not limited to biological and medicalsystems. Their usage includes chemical reaction sys-tems, queuing systems, ecological systems, economicsystems, telecommunication systems, transportationsystems, and power systems, to cite but a few ex-amples. A key physical limitation of such systems isthat transfers between compartments are not instanta-neous and realistic models for capturing the dynamicsof such systems should account for material, en-ergy, or information in transit between compartments[8,12,16]. Hence, to accurately describe the evolu-tion of the aforementioned systems, it is necessary

0167-6911/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2003.09.006

356 W.M. Haddad, V. Chellaboina / Systems & Control Letters 51 (2004) 355–361

to include in any mathematical model of the systemdynamics some information of the past system states.This of course leads to (in8nite-dimensional) delaydynamical systems [11,19].In this paper we develop necessary and su7cient

conditions for time-delay nonnegative and compart-mental dynamical systems. Speci8cally, using linearLyapunov–Krasovskii functionals we develop nec-essary and su7cient conditions for asymptotic sta-bility of linear nonnegative dynamical systems withtime delay. The consideration of a linear Lyapunov–Krasovskii functional leads to a new Lyapunov-likeequation for examining stability of time delay non-negative dynamical systems. The motivation for us-ing a linear Lyapunov–Krasovskii functional followsfrom the fact that the (in8nite-dimensional) stateof a retarded nonnegative dynamical system is non-negative and hence a linear Lyapunov–Krasovskiifunctional is a valid candidate Lyapunov–Krasovskiifunctional. For a time delay compartmental system,a linear Lyapunov–Krasovskii functional is shown tocorrespond to the total mass of the system at a giventime plus the integral of the mass Fow in transit be-tween compartments over the time intervals it takesfor the mass to Fow through the intercompartmentalconnections. It is important to note that even thoughlinear Lyapunov functions have been considered in[13,17] for compartmental systems, linear Lyapunovfunctionals for linear and nonlinear compartmentaland nonnegative dynamical systems with time delayshave not been considered in the literature.

2. Notation and mathematical preliminaries

In this section, we introduce notation, severalde8nitions, and some key results concerning linearnonnegative dynamical systems [3,4,9] that are nec-essary for developing the main results of this paper.Speci8cally, N denotes the set of nonnegative inte-gers, R denotes the reals, and Rn is an n-dimensionallinear vector space over the reals with the maximummodulus norm ‖ · ‖ given by ‖x‖ = maxi=1; :::; n |xi|,x∈Rn. For x∈Rn, we write x¿¿ 0 (resp., x�0)to indicate that every component of x is nonnegative(resp., positive). In this case we say that x is nonneg-ative or positive, respectively. Likewise, A∈Rn×m is

nonnegative 1 or positive if every entry of Ais nonnegative or positive, respectively, whichis written as A¿¿ 0 or A�0, respectively.Let IRn

+ and Rn+ denote the nonnegative and

positive orthants of Rn; that is, if x∈Rn, thenx∈ IRn

+ and x∈Rn+ are equivalent, respectively, to

x¿¿ 0 and x�0. Finally, C([a; b];Rn) denotes aBanach space of continuous functions mapping theinterval [a; b] into Rn with the topology of uni-form convergence. For a given real number ¿ 0if [a; b] = [ − ; 0] we let C = C([ − ; 0];Rn)and designate the norm of an element � in Cby |||�||| = sup�∈[−;0] ‖�(�)‖. If ; �∈R andx∈C([ − ; + �];Rn), then for every t ∈ [ ; + �],we let xt ∈C be de8ned by xt(�)=x(t+�), �∈ [−; 0].The following de8nition introduces the notion of anonnegative function.

De�nition 2.1. Let T ¿ 0. A real function u :[0; T ] → Rm is a nonnegative (resp., positive) func-tion if u(t)¿¿ 0 (resp., u(t)�0) on the interval[0; T ].

The next de8nition introduces the notion of essen-tially nonnegative matrices.

De�nition 2.2 (Berman and Plemmons): Let A∈Rn×n. A is essentially nonnegative if A(i; j)¿ 0,i; j = 1; : : : ; n, i �= j.

Recall that a linear dynamical system

x(t) = Ax(t); x(0) = x0; t¿ 0; (1)

where x(t)∈Rn, t¿ 0, and A∈Rn×n is essentiallynonnegative, is said to be a linear nonnegative dynam-ical system [9]. The solution to (1) is standard and isgiven by x(t)=eAtx(0), t¿ 0. Furthermore, recall that(see Theorem 3 of [2, p. 176]) A is essentially non-negative if and only if the state transition matrix eAt isnonnegative on [0;∞). Hence, if A is essentially non-negative and x0¿¿ 0, then x(t)¿¿ 0, t¿ 0, wherex(t), t¿ 0, denotes the solution to (1).Next, we consider a subclass of nonnegative sys-

tems; namely, compartmental systems.

1 In this paper it is important to distinguish between a squarenonnegative (resp., positive) matrix and a nonnegative-de8nite(resp., positive-de8nite) matrix.

W.M. Haddad, V. Chellaboina / Systems & Control Letters 51 (2004) 355–361 357

De�nition 2.3. Let A∈Rn×n. A is a compartmen-tal matrix if A is essentially nonnegative and∑n

i=1 A(i; j)6 0, j = 1; 2; : : : ; n.

If A is a compartmental matrix, then the nonneg-ative system (1) is called an in6ow-closed compart-mental system [9,12,13]. Recall that an inFow-closedcompartmental system possesses a dissipation prop-erty and hence is Lyapunov stable since the totalmass in the system given by the sum of all compo-nents of the state x(t), t¿ 0, is nonincreasing alongthe forward trajectories of (1). In particular, withV (x) = eTx, where e = [1; 1; : : : ; 1]T, it follows thatV (x) = eTAx =

∑nj=1

[∑ni=1 A(i; j)

]xj6 0, x∈ IRn

+.Furthermore, since ind(A)6 1, where ind(A) de-notes the index of A, it follows that A is semistable;that is, limt→∞ eAt exists. Hence, all solutions ofinFow-closed linear compartmental systems are con-vergent. For details of the above facts, see [4,9].

3. Stability theory for nonnegative dynamicalsystems with time delay

In the 8rst part of this paper we consider a lineartime delay dynamical system G of the form

x(t) = Ax(t) + Adx(t − ); x(�) = �(�);

−6 �6 0; t¿ 0; (2)

where x(t)∈Rn, t¿ 0, A∈Rn×n, Ad ∈Rn×n, ¿ 0,and �(·)∈C=C([− ; 0];Rn) is a continuous vectorvalued function specifying the initial state of the sys-tem. Note that the state of (2) at time t is the pieceof trajectories x between t− and t, or, equivalently,the element xt in the space of continuous functions de-8ned on the interval [− ; 0] and taking values in Rn;that is, xt ∈C([ − ; 0];Rn). Hence, xt(�) = x(t + �),�∈ [− ; 0]. Furthermore, since for a given time t thepiece of the trajectories xt is de8ned on [ − ; 0], theuniform norm |||xt ||| = sup�∈[−;0] ‖x(t + �)‖ is usedfor the de8nitions of Lyapunov and asymptotic stabil-ity of (2). For further details see [11,14]. Finally, notethat since �(·) is continuous it follows from Theorem2.1 of [11, p. 14] that there exists a unique solutionx(�) de8ned on [ − ;∞) that coincides with � on[− ; 0] and satis8es (2) for t¿ 0. The following def-inition is needed for the main results of this section.

De�nition 3.1. The linear time delay dynami-cal system G given by (2) is nonnegative iffor every �(·)∈C+, where C+ , { (·)∈C: (�)¿¿ 0; �∈ [− ; 0]}, the solution x(t), t¿ 0, to (2)is nonnegative.

Proposition 3.1. The linear time delay dynamicalsystem G given by (2) is nonnegative if and only ifA∈Rn×n is essentially nonnegative and Ad ∈Rn×n isnonnegative.

Proof. It follows from Lagrange’s formula that thesolution to (2) is given by

x(t) = eAtx(0) +∫ t

0eA(t−�)Adx(�− ) d�

= eAt�(0) +∫ t−

−eA(t−−�)Adx(�) d�: (3)

Now, the proof follows directly from (3) using thefact that A is essentially nonnegative if and only if eAt

is nonnegative for all t¿ 0.

Remark 3.1. Note that it follows from Proposition3.1 that, since Ad in (2) is required to be nonnegative,the linear time delay dynamical system given by

x(t) = Ax(t − ); x(�) = �(�);

−6 �6 0; t¿ 0; (4)

where x(t)∈Rn, t¿ 0, A∈Rn×n is essentially non-negative, ¿ 0, and�(·)∈C+, fails to yield a nonneg-ative trajectory under arbitrarily small delays ¿ 0.As an example, consider the scalar (n=1) linear timedelay dynamical system G given by x(t)=−x(t− ),x(�) = �(�), −6 �6 0, t¿ 0. Now, note that G isnonnegative if = 0 whereas G is not nonnegative if¿ 0 with �(�) = −�. To see this it need only benoted that x() = x(0) +

∫ 0− −x(�) d�=−2=2.

For the remainder of this section, we assume thatA is essentially nonnegative and Ad is nonnegative sothat for every �(·)∈C+, the linear time delay dynam-ical system G given by (2) is nonnegative. Next, wepresent necessary and su7cient conditions for asymp-totic stability for the linear time delay nonnegativedynamical system (2). Note that for addressing thestability of the zero solution of a time delay nonneg-ative system, the usual stability de8nitions given in

358 W.M. Haddad, V. Chellaboina / Systems & Control Letters 51 (2004) 355–361

[11] need to be slightly modi8ed. In particular, stabil-ity notions for nonnegative dynamical systems needto be de8ned with respect to relatively open subsetsof IRn

+ containing the equilibrium solution xt ≡ 0.For a similar de8nition see [9]. In this case, standardLyapunov–Krasovskii stability theorems for nonlineartime delay systems [11] can be used directly with therequired su7cient conditions veri8ed on C+.

Theorem 3.1. Consider the linear nonnegative timedelay dynamical system G given by (2) whereA∈Rn×n is essentially nonnegative and Ad ∈Rn×n

is nonnegative. Then G is asymptotically stable forall ∈ [0;∞) if and only if there exist p; r ∈Rn suchthat p�0 and r�0 satisfy

0 = (A+ Ad)Tp+ r: (5)

Proof. To prove necessity, assume that the linear timedelay dynamical system G given by (2) is asymptot-ically stable for all ∈ [0;∞). In this case, it followsthat the linear nonnegative dynamical system

x(t) = (A+ Ad)x(t); x(0) = x0 ∈ IRn+; t¿ 0 (6)

or, equivalently, (2) with =0, is asymptotically sta-ble. Now, it follows from the Perron–Frobenius theo-rem [3, p. 27] (see also Theorem 3.2 of [9]) that thereexists p�0 and r�0 such that (5) is satis8ed. Con-versely, to prove su7ciency, assume that (5) holdsand consider the candidate Lyapunov–Krasovskiifunctional V : C+ → R given by

V ( ) = pT (0) +∫ 0

−pTAd (�) d�; (·)∈C+:

Now, note that V ( )¿pT (0)¿ ‖ (0)‖, where , mini∈{1;2; :::; n} pi ¿ 0. Next, using (5), it followsthat the Lyapunov–Krasovskii directional derivativealong the trajectories of (2) is given by

V (xt) =pTx(t) + pTAd[x(t)− x(t − )]

=pT(A+ Ad)x(t) =−rTx(t)6− �‖x(t)‖;where �, mini∈{1;2; :::; n} ri ¿ 0 and xt(�)= x(t+ �),�∈ [−; 0], denotes the (in8nite-dimensional) state ofthe time delay dynamical system G. Now, it followsfrom Corollary 3.1 of [11, p. 143] that the linear non-negative time delay dynamical system G is asymptot-ically stable for all ∈ [0;∞).

Remark 3.2. The results presented in Proposition 3.1and Theorem 3.1 can be easily extended to systemswith multiple delays of the form

x(t) = Ax(t) +nd∑i=1

Adi x(t − i); x(�) = �(�);

− I6 �6 0; t¿ 0; (7)

where x(t)∈Rn, t¿ 0, A∈Rn×n is essentially non-negative, Adi ∈Rn×n, i=1; : : : ; nd, is nonnegative, I=maxi∈{1; :::; nd}i, and �(·)∈{ (·)∈C([ − I; 0];Rn) : (�)¿¿ 0; �∈ [− I; 0]}. In this case, (5) becomes

0 =

(A+

nd∑i=1

Adi

)Tp+ r; (8)

which is associated with the Lyapunov–Krasovskiifunctional

V ( ) = pT (0) +nd∑i=1

∫ 0

−ipTAdi (�) d�: (9)

Remark 3.3. An analogous theorem to Theorem 3.1can be derived for discrete-time, nonnegative time de-lay dynamical systems G of the form

x(k + 1) = Ax(k) + Adx(k − �); x(�) = �(�);

−�6 �6 0; k ∈N; (10)

where x(k)∈Rn, k ∈N, A∈Rn×n and Ad ∈Rn×n

are nonnegative, �∈N, and �(·)∈C+, whereC+ , { (·)∈C({−�; : : : ; 0};Rn): (�)¿¿ 0;�∈{−�; : : : ; 0}} is a vector sequence specifying theinitial state of the system. In this case, G is asymptot-ically stable for all �∈N if and only if there existp; r ∈Rn such that p�0 and r�0 satisfy

p= (A+ Ad)Tp+ r: (11)

The Lyapunov–Krasovskii functional V : C+ → Rused to prove this result is given by

V ( ) = pT (0) +−1∑

�=−�

pTAd (�); (·)∈C+:

(12)

A similar remark holds for Theorem 3.2 below.

Next, we show that inFow-closed, linear compart-mental dynamical systems with time delays [12] are aspecial case of the linear nonnegative time delay sys-tems (2). To see this, for i = 1; : : : ; n, let xi(t), t¿ 0,

W.M. Haddad, V. Chellaboina / Systems & Control Letters 51 (2004) 355–361 359

Fig. 1. Linear compartmental interconnected subsystem model withtime delay.

denote the mass and (hence a nonnegative quantity) ofthe ith subsystem of the compartmental system shownin Fig. 1, let aii¿ 0 denote the loss coe7cient of theith subsystem, and let �ij(t− ), i �= j, denote the netmass Fow (or Fux) from the jth subsystem to the ithsubsystem given by �ij(t− )=aijxj(t− )−ajixi(t),where the transfer coe7cient aij¿ 0, i �= j, and isthe 8xed time it takes for the mass to Fow from thejth subsystem to the ith subsystem. For simplicity ofexposition we have assumed that all transfer times be-tween compartments are given by . The more gen-eral multiple delay case can be addressed as shownin Remark 3.1. Now, a mass balance for the wholecompartmental system yields

xi(t) =−aii+

n∑j=1; i �=j

aji

xi(t)+

n∑j=1; i �=j

aijxj(t−);

t¿ 0; i = 1; : : : ; n; (13)

or, equivalently,

x(t) = Ax(t) + Adx(t − ); x(�) = �(�);

−6 �6 0; t¿ 0; (14)

where x(t) = [x1(t); : : : ; xn(t)]T, �(·)∈C+, and fori; j = 1; : : : ; n,

A(i; j) =

−n∑

k=1

aki; i = j;

0; i �= j;

Ad(i; j) =

{0; i = j;

aij; i �= j:(15)

Note that A is essentially nonnegative and Ad is non-negative. Furthermore, A+Ad is a compartmental ma-trix and hence it follows from Lemma 2.2 of [4] thatRe�¡ 0 or �=0, where � is an eigenvalue of A+Ad.

Now, it follows from Theorem 3.2 of [9] and The-orem 3.1 that the zero solution x(t) ≡ 0 to (14) isasymptotically stable for all ∈ [0;∞) if and only ifA+ Ad is Hurwitz. Alternatively, asymptotic stabilityof (14) for all ∈ [0;∞) can be deduced using theLyapunov–Krasovskii functional

V ( ) = eT (0) +∫ 0

−eTAd (�) d�; (·)∈C+;

(16)

which captures the total mass of the system at t = 0plus the integral of the mass Fow in transit betweencompartments over the intervals it takes for the massto Fow through the intercompartmental connections.In this case, it follows that V (xt)6− �‖x(t)‖, where�, mini∈{1; :::; n} aii and xt(�)=x(t+�), �∈ [−; 0].This result is not surprising, since for an inFow-closed compartmental system the law of conservationof mass eliminates the possibility of unboundedsolutions.Next, we present a nonlinear extension of Proposi-

tion 3.1 and Theorem 3.1. Speci8cally, we considernonlinear time delay dynamical systems G of the form

x(t) = Ax(t) + fd(x(t − )); x(�) = �(�);

−6 �6 0; t¿ 0; (17)

where x(t)∈Rn, t¿ 0, A∈Rn×n, fd : Rn → Rn islocally Lipschitz and fd(0) = 0, ¿ 0, and �(·)∈C.Once again, since �(·) is continuous, existence anduniqueness of solutions to (17) follow from Theorem2.3 of [11, p. 44]. Nonlinear time delay systems of theform given by (17) arise in the study of physiologicaland biomedical systems [5], ecological systems [7],population dynamics [15], as well as neural Hop8eldnetworks [18]. For the nonlinear time delay dynami-cal system (17), the de8nition of nonnegativity holdswith (2) replaced by (17). The following de8nition isneeded for our next result.

De�nition 3.2. Let fd = [fd1; : : : ; fdn]T : D → Rn,where D is an open subset of Rn that contains IRn

+.Then fd is nonnegative if fdi(x)¿ 0, for all i =1; : : : ; n, and x∈ IRn

+.

Proposition 3.2. Consider the nonlinear time delaydynamical system G given by (17). If A∈Rn×n is es-sentially nonnegative and fd : Rn → Rn is nonnega-tive, then G is nonnegative.

360 W.M. Haddad, V. Chellaboina / Systems & Control Letters 51 (2004) 355–361

Next, we present su7cient conditions for asymp-totic stability for nonlinear nonnegative dynamicalsystems given by (17).

Theorem 3.2. Consider the nonlinear nonnegativetime delay dynamical system G given (17) whereA∈Rn×n is essentially nonnegative, fd : Rn → Rn

is nonnegative, and fd(x)66 �x, x∈ IRn+, where

�¿ 0. If there exist p; r ∈Rn such that p�0 andr�0 satisfy

0 = (A+ �In)Tp+ r; (18)

then G is asymptotically stable for all ∈ [0;∞).

Proof. Assume that (18) holds and consider the can-didate Lyapunov–Krasovskii functional V : C+ → Rgiven by

V ( ) = pT (0) +∫ 0

−pTfd( (�)) d�; (·)∈C+:

Now, note that V ( )¿pT (0)¿ ‖ (0)‖, where , mini∈{1;2; :::; n} pi ¿ 0. Next, using (18), it followsthat the Lyapunov–Krasovskii directional derivativealong the trajectories of (17) is given by

V (xt) = pTx(t) + pT[fd(x(t))− fd(x(t − ))]

= pT(Ax(t) + fd(x(t)))

6pTAx(t) + �pTx(t) =−rTx(t)6− �‖x(t)‖;

where � , mini∈{1;2; :::; n} ri ¿ 0. Now, it followsfrom Corollary 3.1 of [11, p. 143] that the nonlin-ear nonnegative time delay dynamical system G isasymptotically stable for all ∈ [0;∞).

Remark 3.4. The structural constraintfd(x)66 �x,x∈ IRn

+, where �¿ 0, in the statement of Theorem 3.2is naturally satis8ed for many compartmental dynam-ical systems. For example, in nonlinear pharmacoki-netic models [10] the transport across biological mem-branes may be facilitated by carrier molecules withthe Fux described by a saturable from fdi(xi; xj) =�max[x i =(x

i + �)− x j =(x

j + �)], where xi, xj are the

concentrations of the ith and jth compartments and�max, , and � are model parameters.

4. Conclusion

Nonnegative and compartmental models are widelyused to capture system dynamics involving the in-terchange of mass and energy between homogeneoussubsystems or compartments. In certain applicationsof these systems, transfers between compartments arenot instantaneous and hence realistic models for thesesystems should account for material or energy in tran-sit between compartments. In this paper, necessary andsu7cient conditions for asymptotic stability of non-negative dynamical systems with time delay are given.

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