haddad.gatech.edu · 36 w.m. haddad, v.s. chellaboina / nonlinear analysis:real world applications...

Nonlinear Analysis: Real World Applications 6 (2005) 35–65

www.elsevier.com/locate/na

Stability and dissipativity theory for nonnegativedynamical systems: a unified analysis frameworkfor biological and physiological systems�

Wassim M. Haddada,∗, VijaySekhar ChellaboinabaSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USAbMechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

Received 11 February 2003; accepted 24 January 2004

Abstract

Nonnegative dynamical system models are derived from mass and energy balance considerationsthat involve dynamic states whose values are nonnegative. Thesemodels are widespread in biological,physiological, and ecological sciences and play a key role in the understanding of these processes.In this paper we develop several results on stability, dissipativity, and stability of feedback intercon-nections of linear and nonlinear nonnegative dynamical systems. Specifically, usinglinear Lyapunovfunctions we develop necessary and sufficient conditions for Lyapunov stability, semistability, thatis, system trajectory convergence to Lyapunov stable equilibrium points, and asymptotic stabilityfor nonnegative dynamical systems. In addition, usinglinear andnonlinearstorage functions withlinearsupply rateswe develop new notions of dissipativity theory for nonnegative dynamical systems.Finally, these results are used to develop general stability criteria for feedback interconnections ofnonnegative dynamical systems.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Nonnegative systems; Compartmental models; Stability theory; Dissipativity theory; LinearLyapunov functions; Storage functions; Supply rates

� This research was supported in part by the National Science Foundation under Grants ECS-9496249 andECS-0133038, and the Air Force Office of Scientific Research under Grant F49620-03-1-0178.

∗ Corresponding author. Tel.: +1-404-894-1078; fax: +1-404-894-2760.E-mail addresses: [email protected](W.M. Haddad), [email protected]

(V.S. Chellaboina).

1468-1218/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2004.01.006

http://www.elsevier.com/locate/na

mailto:[email protected]

mailto:[email protected]

36 W.M. Haddad, V.S. Chellaboina / Nonlinear Analysis:Real World Applications 6 (2005) 35–65

1. Introduction

With the mergence of engineering disciplines and biological and medical sciences, it isnot surprising that dynamical systems theory has played a central role in the understandingof biological and physiological processes[32]. With this unification it has rapidly becomeapparent that mathematical modeling and dynamical systems theory is the key thread thatties together these diverse disciplines. The dynamical models of many biological and phys-iological processes such as pharmacokinetics[3,48], metabolic systems[13], epidemicdynamics[28], biochemical reactions[15,32], endocrine systems[13], and lipoprotein ki-netics[28] are derived from mass and energy balance considerations that involve dynamicstates whose values are nonnegative. Hence, it follows from physical considerations thatthe state trajectory of such systems remains in the nonnegative orthant of the state spacefor nonnegative initial conditions. Such systems are commonly referred to asnonnegativedynamical systemsin the literature[4,16,30,31,40,41,43]. A subclass of nonnegative dy-namical systemsarecompartmental systems[2,8,21,22,28,29,36,37,39,44].Compartmentalsystems involve dynamical models that are characterized by conservation laws (e.g., massand energy) capturing the exchange of material between coupled macroscopic subsystemsknown as compartments. Each compartment is assumed to be kinetically homogeneous;that is, any material entering the compartment is instantaneously mixed with the material ofthe compartment. Since biological and physiological systems have numerous input–outputproperties related to conservation, dissipation, and transport of mass and energy, nonneg-ative and compartmental models are conceptually simple yet remarkably effective in de-scribing the essential phenomenological features of these dynamical systems. Furthermore,since such systems are governed by conservation laws and are comprised of homogeneouscompartments which exchange variable nonnegative quantities of material via intercom-partmental flow laws, these systems are completely analogous to network thermodynamic(reaction–diffusion) systems with compartmental masses (or concentrations) playing therole of temperatures. The range of applications of nonnegative and compartmental systemsis not limited to biological and medical systems. Their usage includes chemical reactionsystems[7,17], queuing systems, large-scale systems[46,47], stochastic systems (whosestate variables represent probabilities)[51], ecological systems[37], and economic systems[5], to cite but a few examples.In this paper we develop several basic mathematical results on stability, dissipativity,

and stability of feedback interconnections of linear and nonlinear nonnegative dynami-cal systems. Specifically, usinglinear Lyapunov functions, we first develop necessary andsufficient conditions for Lyapunov stability, semistability; that is, system trajectory con-vergence to Lyapunov stable equilibrium points[9,12], and asymptotic stability for linearnonnegative dynamical systems. The consideration of a linear Lyapunov function leads to anewLyapunov-like equation for examining the stability of linear nonnegative systems. ThisLyapunov-like equation is analyzed using nonnegativematrix theory[5,27]. Themotivationfor using a linear Lyapunov function follows from the fact that the state of a nonnegativedynamical system is nonnegative and hence a linear Lyapunov function is a valid Lyapunovfunction candidate. This considerably simplifies the stability analysis of nonnegative dy-namical systems. Linear Lyapunov functions were considered in[29] for compartmentalsystems and further explored in[7] to study the stability of mass action kinetics which

W.M. Haddad, V.S. Chellaboina / Nonlinear Analysis:Real World Applications 6 (2005) 35–6537

exhibit nonnegative dynamics. For compartmental systems, a linear Lyapunov functioncorresponds to the total mass of the system. Polytopic Lyapunov functions, which includelinear Lyapunov functions, are addressed in[11,42].It iswell known that linear Lyapunov stable compartmental systemsare semistable[8,29],

however, this is not the case for nonlinear compartmental systems. In fact, nonlinear com-partmental systems can exhibit limit cycles, bifurcations, and even chaos[29]. Hence, it is ofinterest to determinenecessary and sufficient conditions underwhichmasses/concentrationsfor nonlinear compartmental systems converge. Even though sufficient conditions do existin the literature for the absence of limit cycles of nonlinear compartmental systems, theyare extremely conservative requiring the Jacobian of the compartmental system to be co-operative in the nonnegative orthant of the state space for all system states[26,37,44]. Inthis paper we develop necessary and sufficient conditions for identifying nonnegative andcompartmental systems that only admit monotonic solutions. As a result, we providenewsufficient conditions for the absence of limit cycles in nonlinear compartmental systems.Furthermore, we develop global existence and uniqueness of solution results for nonlinearcompartmental systems.Next, usinglinear andnonlinearstorage functions withlinear supply rates we develop

newnotions of classical dissipativity theory[49,50] and exponential dissipativity theory[14] for nonnegative dynamical systems. The overall approach provides a new interpre-tation of a mass balance for nonnegative systems with linear supply rates and linear andnonlinear storage functions. Specifically, we show that dissipativity of a nonnegative dy-namical system involving a linear storage function and a linear supply rate implies that thesystem mass transport is equal to the supplied system flux minus the expelled system flux.In the special case where the linear supply rate is taken to be the excess input mass fluxover the output mass flux the system dissipativity notion collapses to anonaccumulativitysystem constraint wherein the system mass transport is always less than or equal to thedifference between the system flux input and system flux output. Furthermore, we showthat all compartmental systems with measured outputs corresponding to material outflowsare nonaccumulative. In addition, we developnewKalman–Yakubovich–Popov equationsfor nonnegative systems for characterizing dissipativeness with linear and nonlinear storagefunctions and linear supply rates. Using the concepts of dissipativity and exponential dissi-pativity for nonnegative dynamical systems, we develop feedback interconnection stabilityresults for nonnegative systems. General stability criteria are given for Lyapunov, semi,and asymptotic stability of feedback nonnegative dynamical systems. These results can beviewed as a generalization of the positivity and the small gain theorems[25] to nonnegativesystems with linear supply rates involving net input–output system mass flux. Finally, westress that although some of the stability results onlinearnonnegative dynamical systems inthis paper are well known[16,31], they are stated here in a concise and unified format thatsupports the development of dissipativity theory for nonnegative systems in later sections.

2. Mathematical preliminaries

In this section we introduce several definitions and some key results concerning nonnega-tive matrices[5,6,18,27,38]that are necessary for developing the main results of this paper.


Definition 2.1. LetA ∈ Rm×n. ThenA isnonnegative1 (resp.,positive) if A(i,j)�0 (resp.,A(i,j) >0) for all i = 1,2, . . . , m andj = 1,2, . . . , n.

Definition 2.2. Let T >0. A real functionu : [0, T ] → Rm is a nonnegative(resp.,positive) functionif u(t)� �0 (resp.,u(t)?0) on the interval[0, T ].

Definition 2.3 (Berman and Plemmons[5] ). LetA ∈ Rn×n.A is aZ-matrix if A(i,j)�0,i, j = 1, . . . , n, i = j . A is anM-matrix (resp., anonsingularM-matrix) if A is aZ-matrix and all the principal minors ofA are nonnegative (resp., positive).A is essentiallynonnegativeif −A is aZ-matrix; that is,A(i,j)�0, i, j = 1, . . . , n, i = j .

The following lemma is needed for developing several stability results in later sections.

Lemma 2.1. AssumeA ∈ Rn×n is a Z-matrix. Then the following statements areequivalent:

(i) A is anM-matrix.(ii) There exists a scalar�>0and ann×n nonnegativematrixB� �0such that��(B)

andA = �I − B.(iii) Re��0, � ∈ spec(A).(iv) If � ∈ spec(A), then either� = 0 or Re �>0.Furthermore, in the case whereA ∈ Rn×n is a nonsingularZ-matrix, then the following

statements are equivalent:

(v) A is a nonsingularM-matrix.(vi) detA = 0 andA−1� �0.(vii) For eachy ∈ Rn, y� �0, there exists a uniquex ∈ Rn, x� �0, such thatAx = y.(viii) There existsx ∈ Rn, x� �0, such thatAx?0.(ix) There existsx ∈ Rn, x?0, such thatAx?0.

Proof. The equivalence of statements (i)–(ii) follows from[27, p. 119]. The equivalence ofstatements (i)–(iv) follows from[5, p. 150]. The equivalence of statements (v), (vi), (viii),and (ix) follows from Theorem 6.2.3 of[5]. Finally, (vi) implies (vii) is immediate while(vii) implies (viii) follows by settingy = e. �

In the first part of this paper we consider linear dynamical systems of the form

x(t) = Ax(t), x(0) = x0, t�0, (1)

wherex(t) ∈ Rn, t�0, andA ∈ Rn×n. The solution to (1) is standard and is givenby x(t) = eAtx(0), t�0. The following lemma proven in[8] shows thatA is essentiallynonnegative, if and only if, the state transition matrix eAt is nonnegative on[0,∞).

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


Lemma 2.2. LetA ∈ Rn×n.ThenA is essentially nonnegative if and only ifeAt is nonnega-tive for all t�0.Furthermore, if A is essentially nonnegative andx0� �0, thenx(t)� �0,t�0,wherex(t), t�0,denotes the solution to(1).

3. Stability theory for linear nonnegative dynamical systems

Linear nonnegative dynamical systems are of major importance in biological and phys-iological systems. For example, almost the entire field of distribution of tracer labelledmaterials in steady state systems can be captured by linear nonnegative dynamical systems[28]. In this section we provide necessary and sufficient conditions for stability of linearnonnegative dynamical systems. Since it follows from Lemma 2.2 that the linear dynamicalsystem given by (1) is nonnegative, if and only if,A is essentially nonnegative, henceforthwe assume thatA is essentially nonnegative. The following definition introduces severaltypes of stability notions corresponding to the equilibrium solutionx(t) ≡ xe of (1), wherexe ∈ N(A), and whose solutions remain in the nonnegative orthantR

n

+.

Definition 3.1. The equilibrium solutionx(t) ≡ xe of the nonnegative dynamical system(1) is Lyapunov stableif, for every ε >0, there exists� = �(ε)>0 such that ifx0 ∈B�(xe) ∩ R

n

+, thenx(t) ∈ Bε(xe) ∩ Rn

+, t�0. The equilibrium solutionx(t) ≡ xe of thenonnegative dynamical system (1) issemistableif it is Lyapunov stable and there exists�>0 such that ifx0 ∈ B�(xe)∩R

n

+, then limt→∞ x(t) exists and converges to a Lyapunovstable equilibrium point. The equilibrium solutionx(t) ≡ xe of the nonnegative dynamicalsystem (1) isasymptotically stableif it is Lyapunov stable and there exists�>0 such thatif x0 ∈ B�(xe) ∩ R

n

+, then limt→∞x(t) = xe. Finally, the equilibrium solutionx(t) ≡ xeof the nonnegative dynamical system (1) isglobally asymptotically stableif the previousstatement holds for allx0 ∈ Rn.

Remark 3.1. Recall that a matrixA ∈ Rn×n is Lyapunov stable, if and only if, thereexists�>0 such that‖eAt‖< �, t�0; semistable if and only if limt→∞ eAt exists; andasymptotically stable if and only if limt→∞ eAt = 0. Note that ifA is asymptotically stablethenN(A) = {0}.

The following theorem gives several properties of a nonnegative dynamical systemwhena Lyapunov-like equation is satisfied for (1). For the special case of undisturbed compart-mental systems[8,28,29]this result specializes to Lemma 2.2 of[8].

Theorem 3.1. LetA ∈ Rn×n be essentially nonnegative. If there exist vectorsp, r ∈ Rn

such thatp?0 andr� �0 satisfy

0= ATp + r, (2)

then the following properties hold:

(i) −A is an M-matrix.(ii) If � ∈ spec(A), then eitherRe �<0 or� = 0.


Fig. 1. Linear compartmental interconnected subsystem model.

(iii) ind(A)�1.(iv) A is semistable andlim t→∞ eAt = I − AA#� �0.(v) R(A) = N(I − AA#) andN(A) = R(I − AA#).(vi)

∫ t

0 eA�ds = A#(eAt − I ) + (I − AA#)t , t�0.

(vii) A is nonsingular if and only if—A is a nonsingularM-matrix.(viii) If A is nonsingular, then A is asymptotically stable andA−1� �0.

Proof. Since, by (2),−ATp� �0 and−A is aZ-matrix, it follows from Theorem 1 of[6]that−AT and hence−A is anM-matrix. Now, the proof is virtually identical to the proofof Lemma 2.2 given in[8]. �

Remark 3.2. It follows from Lemma 2.2 and (iv) of Theorem 3.1 that limt→∞ x(t)= (I −AA#)x0� �0.Hence, the set of all equilibria of a semistable linear nonnegative dynamicalsystem lie in theN(A) = R(I − AA#).

Next, we show that linear compartmental dynamical systems[2,8,21,22,28,29,36,37,39,44]are a special case of nonnegative dynamical systems. To see this, letxi(t), t�0,i = 1, . . . , n, denote the mass (and hence a nonnegative quantity) of theith subsystem ofthe compartmental system shown inFig. 1, let aii �0 denote the loss coefficient of theithsubsystem, letwi(t)�0, t�0, i = 1, . . . , n, denote the flux (mass inflow) supplied to theith subsystem, and let�ij (t), t�0, i = j , i, j = 1, . . . , n, denote the net mass flow (orflux) from thej th subsystem to theith subsystem given by�ij (t) = aij xj (t) − ajixi(t),t�0, where the transfer coefficientaij �0, i = j , i, j = 1, . . . , n. Hence, a mass balancefor the whole compartmental system yields

xi (t) = −aiixi(t) +n∑

j=1,i =j

�ij (t) + wi(t), t�0, i = 1, . . . , n, (3)

or, equivalently,

x(t) = Ax(t) + w(t), x(0) = x0, t�0, (4)


wherex(t) = [x1(t), . . . , xn(t)]T, w(t) = [w1(t), . . . , wn(t)]T, and fori, j = 1, . . . , n,

A(i,j) ={−∑n

k=1aki, i = j,

aij , i = j.(5)

Note that (5) implies that∑n

i=1A(i,j)�0,j=1, . . . , n. Thus, a compartmental system (withw(t) ≡ 0) satisfiesxi(t)�0 for all t�0 wheneverxj (t)=0 for all j = i andt�0, while anonnegative system (withw(t) ≡ 0) satisfiesxi(t)�0 for all t�0 wheneverxi(t) = 0 andxj (t)�0 for all j = i andt�0. Note thatA is an essentially nonnegative matrix and hencethe compartmental model given by (3) is a nonnegative dynamical system. Furthermore,note thatATe = [−a11, −a22, . . . ,−ann]T, and hence withp = e andr = −ATe� �0 itfollows that (2) is satisfied which implies that the compartmental model given by (3) (withw(t) ≡ 0) is semistable ifA is singular and asymptotically stable ifA is nonsingular. In bothcases,V (x) = eTx = ∑n

i=1xi denoting the total mass of the system serves as a Lyapunovfunction for the undisturbed (w(t) ≡ 0) system (3) withV (x) = ∑n

j=1[∑n

i=1A(i,j)]xj =−∑n

i=1aiixi �0, x ∈ Rn

+. The compartmental system (3) with no inflows; that is,wi(t) ≡0, i = 1, . . . , n, is said to beinflow-closed[28]. Alternatively, if (3) possesses no losses(outflows) it is said to beoutflow-closed[28]. A compartmental system is said to beclosedif it is inflow-closed and outflow-closed. Note that for a closed systemV (x) = 0, x ∈ R

n

+,which shows that the total mass inside a closed system is conserved. Alternatively, in thecase whereaii = 0 andwi(t) = 0, t�0, i=1, . . . , n, it follows that (3) can be equivalentlywritten as

x(t) = [Jn(x(t)) − D(x(t))](

�V�x

x(t)

)T+ w(t), x(0) = x0, t�0, (6)

whereJn(x) is a skew-symmetricmatrix functionwithJn(i,i)(x)=0 andJn(i,j)(x)=aij xj −ajixi , i = j , andD(x) = diag[a11x1, a22x2, . . . , annxn]� �0, x ∈ R

n

+. Hence, a linearcompartmental system is aport-controlled Hamiltonian system[35] with a HamiltonianH(x) = V (x) = eTx representing the total mass in the system,D(x) representing theoutflow dissipation, andw(t) representing the supplied flux to the system. This observationshows that compartmental systems are conservative systems. This will be further elaboratedon in Section 5.Finally,wenote that semistability of an inflow-closed compartmental system is equivalent

to the existence of atrap in the system. A trap[29] is a compartment or a set of compart-ments which is outflow-closed and from which there are no transfers to any compartmentsoutside the trap. Asimple-trapis a trap that has no traps inside it. Ref.[19] shows that thecompartmental matrixgiven by (5) has a zero eigenvalue if and only if the compartmentalsystem has a trap. Furthermore,[20] shows that the algebraic multiplicity of� = 0, where� ∈ spec(A), corresponds to the number of simple traps in the system. Since by (iii) ofTheorem3.1 ind(A)�1, it follows that the algebraicmultiplicity and geometricmultiplicityof � ∈ spec(A) (including � = 0) are equal. Hence, all solutions of inflow-closed linearcompartmental systems are convergent and hence semistable.Next, motivated by the fact that for a compartmental system the total mass in the system

can serve as a Lyapunov function, we give necessary and sufficient conditions for Lyapunovstability, semistability, and asymptotic stability for linearnonnegativedynamical systems


usinglinear Lyapunov functions. Even thoughsomeof the results presented below followfrom nonnegative matrix theory[5], here we provide a proof based on standard Lyapunovtheory and invariant set arguments using the Lyapunov-like equation (2).

Theorem 3.2. Consider the linear dynamical system(1) whereA ∈ Rn×n is essentiallynonnegative. Then the following statements hold:

(i) A is Lyapunov stable if and only if A is semistable.(ii) If there exist vectorsp, r ∈ Rn such thatp?0 and r� �0 satisfy(2), then A is

semistable(and hence Lyapunov stable).(iii) If A is semistable, then there exists vectorsp, r ∈ Rn such thatp� �0,p = 0, and

r� �0 satisfy(2).(iv) If there exist vectorsp, r ∈ Rn such thatp� �0 andr� �0 satisfy(2) and(A, rT)

is observable, thenp?0 and(1) is asymptotically stable.Furthermore, the following statements are equivalent:

(v) A is asymptotically stable.(vi) There exist vectorsp, r ∈ Rn such thatp?0 andr?0 satisfy(2).(vii) There exist vectorsp, r ∈ Rn such thatp� �0 andr?0 satisfy(2).(viii) For everyr ∈ Rn such thatr?0, there existsp ∈ Rn such thatp?0 satisfies(2).

Proof. (i) If A is semistable thenA is Lyapunov stable by definition. Conversely, supposeA is Lyapunov stable and essentially nonnegative. Thus, it follows from (ii) of Lemma 2.1that−A is anM-matrix. Now, it follows from (iv) of Lemma 2.1 that the real part of eachnonzero� ∈ spec(A) is negative and hence Re�<0 or� = 0. This proves the equivalencebetween Lyapunov stability and semistability.(ii) The proof is a direct consequence of (iv) of Theorem 3.1. Alternatively, consider the

linear Lyapunov function candidateV (x)=pTx. Note thatV (0)=0 andV (x)>0,x ∈ Rn

+,x = 0. Now, computing the Lyapunov derivative yields

V (x)�=V ′(x)Ax = pTAx = −rTx�0, x ∈ R

n

+,

establishing Lyapunov stability. Semistability now follows from (i).(iii) If A is semistable it follows as in the proof of (i) that−AT is anM-matrix. Hence,

it follows from (ii) of Lemma 2.1 that there exist a scalar�>0 and a nonnegative matrixB� �0 such that��(B) andAT = B − �I . Now, sinceB� �0, it follows from [5]that�(B) ∈ spec(B) and hence there existsp� �0,p = 0, such thatBp = �(B)p. Thus,ATp = Bp − �p = (�(B) − �)p� �0 which proves that there existp� �0,p = 0, andr� �0 such that (2) holds.(iv) Assume there existp� �0 andr� �0 such that (2) holds and suppose(A, rT) is

observable. Now, consider the functionV (x)=pTx, x ∈ Rn

+, and note that sinceV (x)�0andV (x)=pTAx = −rTx�0,x ∈ R

n

+, it follows that ifx(0) ∈ P�={x ∈ R

n

+: pTx = 0},thenV (x(t))=0,t�0,which implies that dV (x(t))/dt=0.Specifically, dV (x(t))/dt |t=0=pTAx(0) = 0. Hence, ifx ∈ P then V (x) = pTAx = −rTx = 0. Thus, if x ∈ P then

Ax ∈ Pandx ∈ Q�={x ∈ R

n

+: rTx = 0}. Now, sinceAx ∈ P it follows thatA2x ∈ P and


Ax ∈ Q. Repeating these arguments yieldsAkx ∈ Q, k = 0,1, . . . , n − 1, or, equivalently,rTAkx = 0, k = 1,2, . . . , n − 1. Now, since(A, rT) is observable it follows thatx = 0andP = {0} which implies thatp?0. Asymptotic stability of (1) now follows as a directconsequence of LaSalle’s invariant set theorem[34] with V (x) = pTx and using the factthat(A, rT) is observable.To show the equivalence between (v)–(viii) first suppose there existsp� �0 andr?0

such that (2) holds. Now, there exists sufficiently smallε >0 such thatAT(p + εe)>0 andp+εe?0which proves that (vii) implies (vi). Since (vi) implies (vii) it trivially follows that(vi) and (vii) are equivalent. Now, suppose (vi) holds; that is, there existsp?0 andr?0

such that (2) holds and consider the Lyapunov function candidateV (x) = pTx, x ∈ Rn

+.Computing the Lyapunov derivative yieldsV (x)=pTAx=−rTx <0,x ∈ R

n

+, x = 0, andhence it follows that (1) is asymptotically stable. Thus, (vi) implies (v). Next, suppose (1) is

asymptotically stable. Hence,−A−T� �0 and thus for everyr ∈ Rn+, p�= −A−Tr� �0

satisfies (2) which proves that (v) implies (vii).Finally, suppose (1) is asymptotically stable. Now, as in the proof given above, for every

r ∈ Rn+, there existsp ∈ Rn

+ such that (2) holds. Next, suppose,ad absurdum, there exists

x ∈ Rn

+, x = 0, such thatxTp = 0; that is, there is at least onei ∈ {1,2, . . . , n}, such thatpi =0. Hence,−xTA−Tr=0. However, since−AT� �0 it follows that−A−1x� �0 and,sincer?0, it follows that−A−1x = 0 which implies thatx = 0 yielding a contradiction.Hence, for everyr ∈ Rn+, there existsp ∈ Rn+ such that (2) holds which proves (v) implies(viii). Since (viii) implies (vi) trivially, the equivalence of (v)–(viii) is established.�

Next, using Theorem 3.2, we show that every asymptotically stable linear nonnegativesystem is equivalent, modulo a similarity transformation, to a compartmental system. Wenote that this result is well known (see for example[16, p. 147]), however, here we give anewproof of this result based on the Lyapunov-like equation (2).

Proposition 3.1. Let A ∈ Rn×n be essentially nonnegative and asymptotically stable.Then there exists a diagonal invertible matrixS ∈ Rn×n such thatA(i,j)�0, i = j , and∑n

k=1A(k,j)�0, i, j = 1, . . . , n, whereA = SAS−1.

Proof. It follows from (v) and (vi) of Theorem 3.2 that there existsp ∈ Rn+ such that

ATp>0.Now, defineS�=diag[p1, . . . , pn], wherepi is theith component ofp. Next, since

A(i,j)=piA(i,j)p−1j , i, j =1, . . . , n, it follows thatA(i,j)�0, i, j =1, . . . , n. Furthermore,

sinceSATe = ATSe = ATp>0 which, sinceS?0 and diagonal, implies thatATe>0.

Hence,∑n

k=1A(k,j)�0, i, j = 1, . . . , n. �

Finally, we give necessary and sufficient conditions for asymptotic stability of a linearnonnegative dynamical system usingquadraticLyapunov functions. The first result appearsin [16,31]and is stated here for completeness.

Theorem 3.3. Consider the dynamical system(1)whereA ∈ Rn×n is essentially nonneg-ative. Then(1) is asymptotically stable if and only if there exists a positive diagonal matrix


P ∈ Rn×n and ann × n positive-definite matrixR such that

0= ATP + PA + R. (7)

Proof. See[16, p. 41]. �

Theorem 3.4. Consider the dynamical system(1)whereA ∈ Rn×n is essentially nonnega-tive. Then(1) is asymptotically stable if and only if for every positive, positive-definiten×n

matrixR, there exists a positive, positive-definiten × n matrixP satisfying(7).

Proof. Sufficiency is immediate. To show necessity, assumeA is essentially nonnegativeand asymptotically stable and letR be a positive, positive definitematrix. SinceA is asymp-totically stable it follows that there exists a uniquen×n positive-definitematrixP such that(7) is satisfied and hence(A⊕A)TvecP =−vecR, where vec(·) denotes the column stackingoperator. Next, sinceA is essentially nonnegative and asymptotically stable it follows thatA ⊕ A is essentially nonnegative and asymptotically stable. Now, sinceR?0, it followsthat vecR?0 and hence (viii) of Theorem 3.2 implies that vecP?0 which establishes thatP?0. �

4. Stability theory for nonlinear nonnegative dynamical systems

Many applications in life sciences give rise to nonlinear nonnegative dynamical systems.These include metabolic pathways, membrane transports, pharmacodynamics, epidemiol-ogy, and ecology[28]. In this section we consider nonlinear dynamical systems of theform

x(t) = f (x(t)), x(0) = x0, t ∈ Ix0, (8)

wherex(t) ∈ D,D is an open subset ofRn with 0 ∈ D, f : D → Rn is locally Lipschitz,andIx0 = [0, �x0), 0< �x0�∞, is the maximal interval of existence for the solutionx(·)of (8). Recall that the pointxe ∈ D is anequilibrium pointof (8) if f (xe)=0. Furthermore,a subsetDc ⊆ D is an invariant setwith respect to (8) ifDc contains the orbits of allits points. Finally, thepositive limit set�(x0) of (8) is the set of points inDc which areapproached along the orbits of (8) with increasing time. The following definition introducesthe notion of essentially nonnegative vector fields[7].

Definition 4.1. Let f = [f1, . . . , fn]T : D → Rn, whereD is an open subset ofRn thatcontainsR

n

+. Thenf isessentially nonnegativeif fi(x)�0, for all i=1, . . . , n, andx ∈ Rn

+such thatxi = 0, wherexi denotes theith element ofx.

Note that iff (x)=Ax, whereA ∈ Rn×n, thenf is essentially nonnegative if and only ifA is essentially nonnegative. The following proposition generalizes Lemma2.2 to nonlinearsystems. This result was first stated in[7].

Proposition 4.1. SupposeRn

+ ⊂ D. ThenRn

+ is an invariant set with respect to(8) if andonly if f : D → Rn is essentially nonnegative.


Proof. Define d(x,Rn

+)�= inf

y∈Rn

+‖x − y‖, x ∈ Rn. Now, supposef : D → Rn is

essentially nonnegative and letx ∈ Rn

+. For everyi ∈ {1, . . . , n} such thatxi =0, it followsthat xi + hf i(x) = hf i(x)�0 for h�0 while for everyi ∈ {1, . . . , n} such thatxi >0,xi + hf i(x)>0 for sufficiently small|h|. Thus,x + hf (x) ∈ R

n

+ for all sufficiently smallh>0 and hence limh→0+d(x +hf (x),R

n

+)/h=0. It now follows from Theorem 4.1.28 of

[1], with x(0)= x, thatx(t) ∈ Rn

+ for all t ∈ Ix0. Conversely, supposex(0)= x ∈ Rn

+ andassume,ad absurdum, that there existsi ∈ {1, . . . , n} such thatxi =0 andfi(x)<0. Then,it follows from continuity that there exists sufficiently smallh>0 such thatfi(x(t))<0for all t ∈ [0, h). Hence,xi(t) is strictly decreasing and hencex(t) /∈ R

n

+ for all t ∈ [0, h)which leads to a contradiction.�

The following result shows that if a nonlinear system is nonnegative then its linearizationis also nonnegative.

Lemma 4.1. Consider the nonlinear dynamical system(8)wheref (0) = 0 andf : D →Rn is essentially nonnegative and continuously differentiable inR

n

+. Then, A�=

�f/�x(x)|x=0 is essentially nonnegative.

Proof. Sincef : D → Rn is essentially nonnegative it follows thatfi(x)|xi=0�0,x ∈ Rn

+.Now, note that for alli = j ,

A(i,j) = �fi�xj

(x)

∣∣∣∣x=0

= limh→0+

fi(0, . . . , h, . . . ,0) − fi(0)

h= lim

h→0+fi(0, . . . , h, . . . ,0)

h,

where h in fi(0, . . . , h, . . . ,0) is on the j th location which implies thatfi(0, . . . ,h, . . . ,0)�0. Hence,A(i,j)�0, i = j , which proves essential nonnegativity ofA. �

Next, we present a key result on stability of a linearized nonlinear nonnegative dynam-ical system. First, however, note that Definition 3.1 also holds for nonlinear nonnegativedynamical systems. In this case, standard Lyapunov stability theorems and invariant settheorems for nonlinear systems[33] can be used directly with the required sufficient condi-tions verified onR

n

+. Furthermore, the definition of a domain of attraction can be extendedto nonlinear nonnegative dynamical systems by restricting the domain to the nonnegativeorthantR

n

+.

Theorem 4.1. Let x(t) ≡ xe be an equilibrium point for the nonlinear dynamical system(8).Furthermore, let f : D → Rn be essentially nonnegative and letA = �f/�x(x)|x=xe.Then the following statements hold:

(i) If Re �<0,where� ∈ spec(A), then the equilibrium solutionx(t) ≡ xe of the nonlin-ear dynamical system(8) is asymptotically stable.

(ii) If there exists� ∈ spec(A) such thatRe �>0, then the equilibrium solutionx(t) ≡ xeof the nonlinear dynamical system(8) is unstable.


(iii) Letxe=0, let Re�<0,where� ∈ spec(A), letp?0 be such thatATp>0,and define

DA�={x ∈ R

n

+: pTx < }, where�= sup{ε >0: pTf (x)<0, x ∈ R

n

+, ‖x‖<ε}and‖x‖ �= ∑n

i=1pi |xi |. ThenDA is a subset of the domain of attraction for(8).

Proof. (i) and (ii) are restatements of Lyapunov’s indirect method[33] as applied to non-linear nonnegative systems. Statement (iii) is a direct consequence of Lemma 4.1 and (vi)of Theorem 3.2. �

Next, we show that nonlinear compartmental dynamical systemsare a special case of non-linear nonnegative dynamical systems. To see this, once again letxi(t), t�0, i = 1, . . . , n,denote the mass (and hence a nonnegative quantity) of theith subsystem of the compart-mental system shown inFig. 1 with aij xj (t) replaced byaij (x(t)) for i, j = 1, . . . , n,let aii(x)�0, x ∈ R

n

+, denote the instantaneous rate of flow of material loss of theithsubsystem, letwi(t)�0, t�0, i = 1, . . . , n, denote the mass inflow supplied to theithsubsystem, and let�ij (x(t)), t�0, i = j , i, j = 1, . . . , n, denote the net mass flow fromthej th subsystem to theith subsystem given by�ij (x(t)) = aij (x(t)) − aji(x(t)), where

the instantaneous rate of material flowaij (x)�0, x ∈ Rn

+, i = j , i, j = 1, . . . , n. Hence,a mass balance for the whole nonlinear compartmental system yields

xi (t) = −aii(x(t)) +n∑

j=1,i =j

[aij (x(t)) − aji(x(t))] + wi(t), t�0, i = 1, . . . , n,

(9)

or, equivalently,

x(t) = f (x(t)) + w(t), x(0) = x0, t�0, (10)

wherex = [x1, . . . , xn]T, w = [w1, . . . , wn]T, f (x) = [f1(x), . . . , fn(x)]T, and fori =1, . . . , n, fi(x) = −aii(x) + ∑n

j=1,i =j [aij (x) − aji(x)]. Since all mass flows as well ascompartment sizes are nonnegative, it follows that for alli = 1, . . . , n, fi(x)�0 for allx ∈ R

n

+ wheneverxi = 0 and whatever the values ofxj , j = i. The above physicalconstraints are implied byaij (x)�0, aii(x)�0, andwi �0, for all i, j = 1, . . . , n, andx ∈ R

n

+; and ifxi =0, thenaii(x)=0 andaji(x)=0 for all i, j =1, . . . , n, andx ∈ Rn

+, sothat xi �0. The above physical constraints imply thatf is essentially nonnegative. Takingthe total mass of the compartmental systemV (x)= eTx = ∑n

i=1xi as a Lyapunov functionfor (10) (withw(t) ≡ 0) and assumingaij (0) = 0, i, j = 1, . . . , n, it follows that

V (x) =n∑

i=1xi

= −n∑

i=1aii(x) +

n∑i=1

n∑j=1,i =j

[aij (x) − aji(x)]

= −n∑

i=1aii(x)�0, x ∈ R

n

+, (11)


which shows that the zero solutionx(t) ≡ 0 of the nonlinear, inflow-closed (w(t) ≡ 0)compartmental system given by (10) is Lyapunov stable. However, unlike the case of linearcompartmental systems, the zero solutionx(t) ≡ 0 to (10) withw(t) ≡ 0 isnotnecessarilysemistable. In fact, (10) can exhibit limit cycles, bifurcations, and chaos (forn�4). Fordetails see[29]. If, however, the Jacobian matrix�f/�x(x) is essentially nonnegative forall x ∈ R

n

+, then it follows fromTheorem 2.1 of[26] (see also[37,44]) that every trajectoryof (10) (withw(t) ≡ 0) is bounded and tends to an equilibrium set; that is, the zero solutionx(t) ≡ 0 of (10) (withw(t) ≡ 0) is convergent. Note that this assumption is automaticallysatisfied for linear nonnegative (and hence linear compartmental) systems.Of course, if (10)withw(t) ≡ 0has losses (outflows) fromall compartments, thenaii(x)>0,x ∈ R

n

+,x = 0,andby (11), the zero solutionx(t) ≡ 0 to (10) (withw(t) ≡ 0) is asymptotically stable.As inthe linear case, nonlinear compartmental systems are port-controlled Hamiltonian systemsand hence conservative systems. This follows from the fact that (9) can be equivalentlywritten as

x(t) = [Jn(x(t)) − D(x(t))](

�V�x

x(t)

)T+ w(t), x(0) = x0, t�0, (12)

whereJn(x) is a skew-symmetricmatrix functionwithJn(i,i)(x)=0andJn(i,j)(x)=aij (x)−aji(x), i = j , andD(x) = diag[a11(x), a22(x), . . . , ann(x)]� �0, x ∈ R

n

+.In light of the above and (12) we have the following result on stability, convergence, and

global existence and uniqueness of solutions for nonlinear inflow-closed compartmentalsystems.

Theorem 4.2. Consider the inflow-closed nonlinear compartmental system(12) whereV (x) = eTx. Then the following statements hold:

(i) If Jn(0) = 0,D(x)� �0, x ∈ Rn

+, andD(0) = 0, then the zero solutionx(t) ≡ 0 to

(12) is Lyapunov stable. If, in addition, D(x)>0, x ∈ Rn

+\{0}, then the zero solutionx(t) ≡ 0 to (12) is asymptotically stable.

(ii) LetDc�={x ∈ R

n

+: V (x)�}be compact for every ∈ R+ and letx0 ∈ Dc.Then thereexists a unique solution to(12) (withw(t) ≡ 0) that is defined for allt�0. In addition,every solutionx(t) → M as t → ∞,whereM is the largest invariant set contained

in R�={x ∈ R

n

+: D(x) = 0}. Finally, if D(i,i)(x) = 0, x ∈ Rn

+, i = 1, . . . , n, exceptfor a finite number of equilibrium points{p1, p2, . . . , pr}, then for everyx0 ∈ Dc,lim t→∞x(t) = pi , wherei ∈ {1, . . . , r}.

Proof. (i) Lyapunovandasymptotic stability followdirectlywithLyapunov functionV (x)=eTx. (ii) Supposex0 ∈ Dc with �V (x0). Now, sinceV (x)�0 for all x ∈ Dc, it fol-lows thatx(t) ∈ Dc for all t�0. Hence, sinceDc is compact, it follows from Theo-rem 2.4 of[33] that there exists a unique solution to (12) that is defined for allt�0.The fact thatx(t) → M as t → ∞ is a direct consequence of LaSalle’s invariantset theorem[33]. Finally, if D(i,i)(x) = 0, x ∈ R

n

+, i = 1, . . . , n, except for a finite

number of equilibrium points{p1, p2, . . . , pr}, thenR = {x ∈ Rn: V (x) = 0} = {x ∈


Rn

+: D(i,i)(x)= 0, i = 1, . . . , n} = {p1, p2, . . . , pr} ∩Dc. Now, it follows that the largestinvariant setM contained inR is given byM = {p1, p2, . . . , pr} ∩ Dc and by LaSalle’sinvariant set theorem[33], x(t) → M = {p1, p2, . . . , pr} ∩ Dc as t → ∞. Next, sincethere exists a unique solution to (12) it follows that for every initial conditionx0 ∈ R

n

+, thepositive limit set�(x0) of (12) is a connected set. Hence, sinceM consists of a set of finiteisolated points it follows thatx(t), t�0, approaches one of the points inM which showsthat limt→∞x(t) = pi , wherei ∈ {1, . . . , r}. �

Remark 4.1. Note that if for every ∈ R+, the set{x ∈ Rn

+: D(x) = 0, eTx = } doesnot contain any closed orbits, thenx(t), t�0, approaches an equilibrium point of (12). Fortwo-dimensional (n=2) nonlinear compartmental systems, since for every ∈ R+, the set{x ∈ R

2+: D(x) = 0, eTx = } is an empty set, a finite set of points, or a line, it follows

that (12) is semistable for everyD(x), x ∈ R2+, andx0 ∈ R

2+.

Finally, we present necessary and sufficient conditions formonotonicityof nonlinearnonnegative dynamical systems. For this result we require the following definition.

Definition 4.2. Consider the nonlinear dynamical system (8) wheref : D → Rn is essen-tially nonnegative. The nonlinear dynamical system (8) ismonotonicif there exists a matrixQ ∈ Rn×n such thatQ = diag[q1, . . . , qn], qi = ±1, i = 1, . . . , n, and for everyx0 ∈ R

n

+,Qx(t2)� �Qx(t1), 0� t1� t2.

Theorem 4.3. The nonlinear dynamical system(8) wheref : D → Rn is essentiallynonnegative andx0 ∈ R

n

+ is monotonic if and only if there exists a matrixQ ∈ Rn×n such

thatQ = diag[q1, . . . , qn], qi = ±1, i = 1, . . . , n, andQf (x)� �0, x ∈ Rn

+.

Proof. To show sufficiency, assume there existsQ=diag[q1, . . . , qn],qi=±1,i=1, . . . , n,such thatQf (x)� �0, x ∈ R

n

+. Now, it follows from (8) that

Qx(t) = Qf (x(t)), x(0) = x0, t�0, (13)

which further implies that

Qx(t2) = Qx(t1) +∫ t2

t1

Qf (x(t))dt. (14)

Next, sincef (·) is essentially nonnegative it follows from Proposition 4.1 thatx(t)� �0,t�0. Hence, sinceQf (x)� �0, x ∈ R

n

+, it follows thatQf (x(t))� �0, t�0, whichimplies that for everyx0 ∈ R

n

+,Qx(t2)� �Qx(t1), 0� t1� t2.To show necessity, assume that (8) is monotonic. In this case, (13) implies that for every

�>0,

Qx(�) = Qx0 +∫ �

0Qf (x(t))dt. (15)

Now, suppose,ad absurdum, there existJ ∈ {1, . . . , n} and x0 ∈ Rn

+ such that[Qf (x0)]J >0. Since the mappingQf (·) and the solutionx(t), t�0, to (8) are contin-


uous it follows that there exists�>0 such that[Qf (x(t))]J >0, 0� t��, which impliesthat[Qx(�)]J > [Qx0]J , leading to a contradiction. Hence,Qf (x)� �0, x ∈ R

n

+. �

As mentioned above, it is of interest to determine sufficient conditions under whichmasses/concentrations for nonlinear compartmental systems are Lyapunov stable and con-vergent, guaranteeing the absence of limit cycling behavior. The following result followsfrom Theorem 4.3 and provides sufficient conditions for the absence of limit cycles innonlinear compartmental systems.

Theorem 4.4. Consider the nonlinear nonnegative dynamical system(12)withw(t) ≡ 0.If there exists a matrixQ ∈ Rn×n such thatQ = diag[q1, . . . , qn], qi = ±1, i = 1, . . . , n,andQf (x)� �0, x ∈ R

n

+, then for everyx0 ∈ Rn

+, limt→∞x(t) exists.

Proof. Let V (x) = eTx, x ∈ Rn

+. Now, it follows from (11) thatV (x(t))�0, t�0, wherex(t), t�0, denotes the solution of (12), which implies thatV (x(t))�V (x0) = eTx0, t�0,and hence for everyx0 ∈ R

n

+, the solutionx(t), t�0, of (11) is bounded. Hence, foreveryi ∈ {1, . . . , n}, xi(t), t�0, is bounded. Furthermore, it follows from Theorem 4.3that xi(t), t�0, is monotonic. Now, sincexi(·), i ∈ {1, . . . , n}, is continuous and everybounded nonincreasing or nondecreasing scalar sequence converges to a finite real number,it follows that limt→∞xi(t), i ∈ {1, . . . , n}, exists. Hence, limt→∞x(t) exists. �

5. Dissipativity theory for nonnegative dynamical systems

In control engineering, dissipativity theory provides a fundamental framework for theanalysis and design of control systems using an input–output description based on systemenergy2 related considerations[35,49,50]. The dissipation hypothesis on dynamical sys-tems results in a fundamental constraint on their dynamic behavior wherein a dissipativedynamical system can only deliver a fraction of its energy to its surroundings and can onlystore a fraction of the work done to it. Such conservation laws are prevalent in dynamicalsystems such as mechanical systems, fluid systems, electrical systems, structural systems,and thermodynamic systems without the consideration of nonnegative and compartmentalmodels. Since biological and physiological systems have numerous input–output propertiesrelated to conservation, dissipation, and transport of mass and energy, it seems natural toextend dissipativity theory to nonnegative and compartmental models which themselvesbehave in accordance to conservation laws. In this section we extend the notion of dissipa-tivity to nonnegative dynamical systems. Specifically, we consider dynamical systemsG ofthe form3

x(t) = f (x(t)) + G(x(t))u(t), x(0) = x0, t�0, (16)

y(t) = h(x(t)) + J (x(t))u(t), (17)

2 Here the notion of energy refers to abstract energy for which a physical system energy interpretation is notnecessary.

3 The outputs here refer tomeasured outputs or observations andmayhave nothing to dowithmaterial outflowsof the nonnegative compartmental system.


wherex ∈ Rn, u ∈ Rm, y ∈ Rl , f : Rn → Rn, G : Rn → Rn×m, h : Rn → Rl , andJ : Rn → Rl×m. We assume thatf (·),G(·), h(·), andJ (·) are continuously differentiablemappings andf (xe)=0 andh(xe)=0. For simplicity of exposition here we assumexe=0.4First, we provide a definition and a key result concerning dynamical systems of the form(16), (17) with nonnegative inputs and nonnegative outputs.

Definition 5.1. The nonlinear dynamical systemG given by (16), (17) isnonnegativeif foreveryx(0) ∈ R

n

+ andu(t)� �0, t�0, the solutionx(t), t�0, to (16) and the outputy(t),t�0, are nonnegative.

Proposition 5.1. Consider the nonlinear dynamical systemG given by(16), (17).If f :D → Rn is essentially nonnegative, h(x)� �0, G(x)� �0, and J (x)� �0, x ∈ R

n

+,thenG is nonnegative.

Proof. The proof is similar to the proof of Proposition 4.1 and hence is omitted.�

Next, we turn our attention to dissipativity theory for nonnegative dynamical systems. Forthe dynamical systemGgiven by (16), (17) a functions : Rm×Rl → R such thats(0,0)=0is called asupply rate[49] if it is locally integrable; that is, for all input–output pairsu ∈ Rm,y ∈ Rl , s(·, ·) satisfies∫ t2

t1|s(u(�), y(�))|d�<∞, t1, t2�0. For the remainder of the results

of this paper we assume thatf (·) is essentially nonnegative,G(x)� �0, h(x)� �0, andJ (x)� �0, x ∈ R

n

+. The following definition introduces the notion of dissipativity for anonnegative dynamical system.

Definition 5.2. The nonnegative dynamical systemG given by (16), (17) isexponentially

dissipative(resp.,dissipative) with respect tothe supply rates : Rm

+ × Rl

+ → R if there

exists a continuous nonnegative-definite functionVs : Rn

+ → R+ called astorage functionand a scalarε >0 (resp.,ε = 0) such thatVs(0) = 0 and thedissipation inequality

eεt2Vs(x(t2))�eεt1Vs(x(t1)) +∫ t2

t1

eεt s(u(t), y(t))dt, t2� t1, (18)

is satisfied for allt1, t2�0, wherex(t), t� t1, is the solution of (16) withu ∈ Rm

+. Thenonnegative dynamical systemG given by (16), (17) islossless with respect to the supply

rate s : Rm

+ × Rl

+ → R if the dissipation inequality (18) is satisfied as an equality withε = 0 for all t2� t1.

Remark 5.1. If Vs(·) is continuously differentiable, then an equivalent statement for ex-ponential dissipativity of a nonnegative dynamical systemG is

Vs(x(t)) + εVs(x(t))�s(u(t), y(t)), t�0, u ∈ Rm

+, y ∈ Rl

+. (19)

4 In the case whereG is nonnegative, this assumption isnot without loss of generality since shifting theequilibrium can destroy the essential nonnegativity of the vector fieldf and the nonnegativity ofh.


Since nonnegative dynamical systems are a subset of dynamical systems, standarddissipativity theory[49,50]with quadratic storage functions andquadratic supply ratesinvolvingKalman–Yakubovich–Popovconditionsalsoholds fornonnegativedynamical sys-tems. In this paper, however,motivated by conservation ofmass laws,we develop dissipativ-ity notions for nonnegative dynamical systems with respect tolinearandnonlinearstoragefunctions andlinear supply rates. The following result presentsnewKalman–Yakubovich–Popov conditions for nonnegative dynamical systems with linear supply rates of the forms(u, y)= qTy + rTu, whereq ∈ Rl , q = 0, andr ∈ Rm, r = 0. To state the main result ofthis section, the following definition is required.

Definition 5.3. A nonnegative dynamical systemG is zero-state observableif for all x ∈R

n

+, u(t) ≡ 0 andy(t) ≡ 0 impliesx(t) ≡ 0. A nonnegative dynamical systemG is

reachableif for all x0 ∈ Rn

+, there exist a finite timeti�0, square integrable inputu(t)defined on[ti,0], such that the statex(t), t� ti , can be driven fromx(ti) = 0 tox(0) = x0.

As in the case of standard dissipativity theory[49], the assumption of reachability guar-antees the existence of a continuous, nonnegative-definite storage functionVs(·) for thenonnegative dynamical systemG. For the remainder of this paper we assume that thereexists a continuously differentiable storage functionVs(x), x ∈ Rn, for the nonnegativedynamical systemG.

Theorem 5.1. Let q ∈ Rl and r ∈ Rm. Consider the nonlinear nonnegative dynamicalsystemG given by(16), (17)wheref : D → Rn is essentially nonnegative, G(x)� �0,h(x)� �0,andJ (x)� �0,x ∈ R

n

+.ThenG is exponentially dissipative(resp., dissipative)with respect to the supply rates(u, y) = qTy + rTu if and only if there exist functionsVs :R

n

+ → R+, , : Rn

+ → R+, andW : Rn

+ → Rm

+, and a scalarε >0 (resp., ε = 0) such

thatVs(·) is continuously differentiable, Vs(0) = 0,and for allx ∈ Rn

+,

0= V ′s(x)f (x) + εVs(x) − qTh(x) + ,(x), (20)

0= V ′s(x)G(x) − qTJ (x) − rT + WT(x). (21)

Proof. Assume that there existVs : Rn

+ → R+, , : Rn

+ → R+, andW : Rn

+ → Rm

+, anda scalarε >0 such thatVs(·) is continuously differentiable,Vs(0) = 0, and (20), (21) hold.Then it follows that for allx ∈ R

n

+ andu ∈ Rm

+,

Vs(x) + εVs(x) = V ′s(x)(f (x) + G(x)u) + εVs(x)

= qTh(x) − ,(x) + qTJ (x)u + rTu − WT(x)u

�qTy + rTu,

which implies thatG is exponentially dissipative with respect to the supply rates(u, y) =qTy+rTu. Conversely, assume thatG is exponentially dissipativewith respect to the supplyrates(u, y) = qTy + rTu. Now, it follows that there exists a continuously differentiablefunctionVs(·) such thatVs(x) + εVs(x) = V ′

s(x)(f (x) + G(x)u) + εVs(x)�qTy + rTu.


Next, letd : Rn

+ × Rm

+ → R+ be such that

d(x, u)�= − Vs(x) − εVs(x) + s(u, y)

= − V ′s(x)(f (x) + G(x)u) − εVs(x)

+ qT(h(x) + J (x)u) + rTu�0, x ∈ Rn

+, u ∈ Rm

+

and note thatd(x, u) is linear in u. Hence, sincex ∈ Rn

+ and u ∈ Rm

+ are arbitrary,

there exist continuous functionsdx : Rn

+ → R+ and du : Rn

+ → R1×m

+ such that

du(x)� �0,x ∈ Rn

+, andd(x, u)=dx(x)+du(x)u, which implies that for allx ∈ Rn

+ andu ∈ R

m

+,

0= V ′s(x)f (x) + εVs(x) − qTh(x) + dx(x)

+ (V ′s(x)G(x) − qTJ (x) − rT + du(x))u.

Now, settingu = 0 yields (20) with,(x) = dx(x) which further implies (21) withW(x) =dTu (x).Finally, the proof of the equivalence between dissipativity with respect to the supply rate

s(u, y) = qTy + rTu and (20) (withε = 0) and (21) follows analogously withε = 0. �

Remark 5.2. As in the standard dissipativity theory with quadratic supply rates[24],the concepts of linear supply rates and linear and nonlinear storage functions provide ageneralized mass and energy interpretation. Specifically, using (20) and (21) it followsthat ∫ t

t0

[qTy(�) + rTu(�)]d� = Vs(x(t)) − Vs(x(t0))

+∫ t

t0

[,T(x(�))x(�) + WT(x(�))u(�)]d�, (22)

which can be interpreted as a generalizedmass balance equation whereVs(x(t))−Vs(x(t0))

is the stored mass of the nonnegative system and the second path-dependent term on theright corresponds to the expelled mass of the nonnegative system. Rewriting (22) as

Vs(x) = V ′s(x)x = qTy + rTu − [,T(x)x + WT(x)u], (23)

yields a mass conservation equation which shows that the systemmass transport is equal tothe supplied system flux minus the expelled system flux.

Remark 5.3. Recall that in standard dissipativity theory ifG is zero-state observable andthere exists a function� : Rl → Rm such thats(�(y), y)<0, y = 0, then the storagefunction Vs(·) satisfiesVs(x)>0, x ∈ Rn, x = 0, [24]. Similarly, for the nonnegativedynamical systemG, it can be shown that ifG is zero-state observable and there exists a

function� : Rl

+ → Rm

+ such thats(�(y), y)<0, y ∈ Rl

+, y = 0, thenVs(x)>0, x ∈ Rn

+,x = 0. Hence, in the case of a linear supply rate, there always exists a matrixK ∈ Rm×l


such thatq + KTr>0 which implies that ifG is zero-state observable, thenVs(x)>0,

x ∈ Rn

+, x = 0.

Remark 5.4. Note that if a nonnegative dynamical systemG is zero-state observable anddissipative with respect to the linear supply rates(u, y)=qTy+rTu, and ifq� �0 andu ≡0, it follows thatVs(x(t))�qTy(t)�0, t�0. Hence, the undisturbed (u(t) ≡ 0) systemG isLyapunovstable.Alternatively, if a nonnegativedynamical systemG is zero-stateobservableand exponentially dissipative with respect to the linear supply rates(u, y) = qTy + rTu,and if q� �0 andu ≡ 0, it follows thatVs(x(t))� − εVs(x(t)) + qTy(t)<0, x(t) = 0,t�0, whereε >0. Hence, the undisturbed (u(t) ≡ 0) systemG is asymptotically stable.

Next, we provide necessary and sufficient conditions for the case whereG given by (16),(17) is lossless with respect to the linear supply rates(u, y) = qTy + rTu.

Theorem 5.2. Let q ∈ Rl and r ∈ Rm. Consider the nonlinear nonnegative dynamicalsystemG given by(16), (17)wheref : D → Rn is essentially nonnegative, G(x)� �0,h(x)� �0, and J (x)� �0, x ∈ R

n

+. ThenG is lossless with respect to the supply rate

s(u, y)= qTy + rTu if and only if there exists a functionVs : Rn

+ → R+ such thatVs(·) iscontinuously differentiable, Vs(0) = 0,and for allx ∈ R

n

+,

0= V ′s(x)f (x) − qTh(x), (24)

0= V ′s(x)G(x) − qTJ (x) − rT. (25)

Proof. The proof is analogous to the proof of Theorem 5.1.�

Next,weprovideakeydefinition fornonnegativedynamical systemswhicharedissipativewith respect to a very special supply rate.

Definition 5.4. A nonnegative dynamical systemG of the form Eqs. (16), (17) isnonac-cumulative(resp.,exponentially nonaccumulative) if G is dissipative (resp., exponentiallydissipative) with respect to the supply rates(u, y) = eTu − eTy.

If G is nonaccumulative, then it follows from (23) that

Vs(x(t))�eTu(t) − eTy(t), t�0, (26)

whereu ∈ Rm

+ andy ∈ Rl

+. If the componentsui(·), i=1, . . . , m, ofu(·) denote flux inputsto the systemG and the componentsyi(·), i = 1, . . . , l, of y(·) denote the flux outputs ofthe systemG, then dissipativity with respect to the linear supply rates(u, y) = eTu − eTy

implies that the systemmass transport is always less than or equal to the difference betweenthe system flux input and system flux output.All compartmental systems with measured outputs corresponding to material outflows

are nonaccumulative. To see this, consider (12)with storage functionVs(x)=eTx and output


y = D(x)(�V/�x)T = [a11(x)x1, a22(x)x2, . . . , ann(x)xn]T. Now, it follows that

Vs(x) = eT

[[Jn(x) − D(x)]

(�V�x

)T+ w

]

= eTw − eTy + eTJn(x)e

= eTw − eTy, x ∈ Rn

+, (27)

which shows that all compartmental systems with outputsy = D(x)(�V/�x)T are loss-less with respect to the supply rates(w, y) = eTw − eTy. If alternatively, the outputsycorrespond to a partial observation of the material outflows, then it follows that the com-partmental system is dissipative with respect to the supply rates(w, y) = eTw − eTy. Tosee this assume without loss of generality that the firstl outflows are observed; that is,y =[a11(x)x1, a22(x)x2, . . . , all(x)xl]. Now, note thaty =D(x)(�V/�x)T −Dr(x)(�V/�x)T,whereDr(x)= diag[0, . . . ,0, al+1l+1(x)xl+1, . . . , ann(x)xn]� �0, x ∈ R

n

+. Hence,

Vs(x) = eT

[[Jn(x) − D(x)]

(�V�x

)T+ w

]

= eTw − eTy −(

�V�x

)TDr(x)

(�V�x

)T�eTw − eTy, x ∈ R

n

+. (28)

Note that in the case where the system is closed,Vs(x)= 0, x ∈ Rn

+, which corresponds toconservation of mass in the system.Finally, we present a key result on linearization of nonnegative dissipative dynamical

systems. For this result we assume that the storage functionVs(·) belongs to C3.

Theorem 5.3. Letq ∈ Rl andr ∈ Rm and consider the nonlinear nonnegative dynamicalsystemG given by(16), (17)wheref : D → Rn is essentially nonnegative, G(x)� �0,h(x)� �0, andJ (x)� �0, x ∈ R

n

+. SupposeG is exponentially dissipative(resp., dis-

sipative) with respect to the supply rates(u, y) = qTy + rTu. Then, there existp ∈ Rn

+,l ∈ R

n

+, andw ∈ Rm

+, and a scalarε >0 (resp., ε = 0) such that

0= ATp + εp − CTq + l, (29)

0= BTp − DTq − r + w, (30)

where

A = �f�x

∣∣∣∣x=0

, B = G(0), C = �h�x

∣∣∣∣x=0

, D = J (0). (31)

If, in addition, (A,C) is observable, thenp?0.

Proof. SinceG is exponentially dissipative (resp., dissipative) with respect to the supplyrates(u, y)= qTy + rTu, it follows that there exists a continuously differentiable function


Vs : Rn

+ → R+ and a scalarε >0 (resp.,ε = 0) such that

Vs(x) + εVs(x) = V ′s(x)[f (x) + G(x)u] + εVs�qTy + rTu, x ∈ R

n

+, u ∈ Rn

+.(32)

Now, it follows from (32) that there exists a functiond : Rn

+ × Rm

+ → R+ such thatd(x, u)�0, d(0,0) = 0, and

0= Vs(x) + εVs(x) − qTy − rTu + d(x, u), x ∈ Rn

+, u ∈ Rn

+. (33)

Next, expandingVs(·) andd(·, ·) via a Taylor series expansion aboutx=0,u=0, and usingthe fact thatVs(·) andd(·, ·) are nonnegative definite andVs(0)= 0, d(0,0)= 0, it followsthat there existsp ∈ R

n

+, l ∈ Rn

+, andw ∈ Rm

+ such that

Vs(x) = pTx + Vsr(x), (34)

d(x, u) = lTx + wTu + dsr(x, u), (35)

whereVsr : Rn

+ → R+ anddsr : Rn

+ × Rm

+ → R+ contain the higher-order terms ofVs(·)andd(·, ·), respectively. Next, letf (x)=Ax+fr(x)andh(x)=Cx+hr(x), wherefr(x)andhr(x), contain the nonlinear terms off (x) andh(x), respectively, and letG(x)=B+Gr(x)

andJ (x)=D + Jr(x), whereGr(x) andJr(x) contain the non-constant terms ofG(x) andJ (x), respectively. Using the above expressions (33) can be written as

0= pTAx + pTBu + εpTx − qT(Cx + Du) − rTu + lTx + wTu + �(x, u), (36)

where�(x, u) is such that�(x, u)/(‖x‖ + ‖u‖) → 0 as‖x‖ + ‖u‖ → ∞. Now, settingu = 0 in (36) and equating coefficients of equal powers yields (29). Alternatively, settingx = 0 in (36) and equating coefficients of equal powers yields (30).Finally, to show thatp?0 in the casewhere(A,C) is observable, note that it follows from

Theorem5.1 that the linearized systemGwith storage functionVs(x)=pTx is exponentiallydissipative (resp., dissipative) with respect to the linear supply rates(u, y) = qTy + rTu.Now, it follows from Remark 5.3 thatp?0. �

6. Specialization to linear nonnegative dynamical systems

In this section we specialize the results of Section 5 to the case of linear nonnegativedynamical systems. Specifically, settingf (x)=Ax,G(x)=B, h(x)=Cx, andJ (x)=D,the nonlinear nonnegative dynamical system given by (16) and (17) specializes to

x(t) = Ax(t) + Bu(t), x(0) = x0, t�0, (37)

y(t) = Cx(t) + Du(t), (38)

wherex ∈ Rn, u ∈ Rm, y ∈ Rl ,A ∈ Rn×n, B ∈ Rn×m,C ∈ Rl×n, andD ∈ Rl×m. Beforeproviding linear dissipativity specializations, we present a key result on linear nonnegativesystems in the case whereu(t)� �0 andy(t)� �0, t�0.

Theorem 6.1. The linear dynamical systemG given by(37), (38)is nonnegative if and onlyif A is essentially nonnegative andB� �0,C� �0,andD� �0.


Proof. See[16, p. 14]. �

The following result presents necessary and sufficient Kalman–Yakubovich–Popov con-ditions for linear nonnegative dynamical systems with linear supply rates of the forms(u, y) = qTy + rTu, whereq ∈ Rl , q = 0, andr ∈ Rm, r = 0. Note that for a lin-ear dynamical system to be dissipative with respect to a linear supply rate it is necessarythat the storage function is also linear. However, since all storage functions are nonnegativeby definition, it follows that a storage function is nonnegative if and only if there exists alinear transformation such that the linear dynamical system is nonnegative in a transformedbasis. Hence, dissipativity theory of linear dynamical systems with respect to linear supplyrates is complete if we restrict our consideration to the class of nonnegative dynamicalsystems.

Theorem 6.2. Letq ∈ Rl andr ∈ Rm.Consider thenonnegativedynamical systemGgivenby(37), (38)whereA is essentially nonnegative,B� �0,C� �0,andD� �0.ThenG isexponentially dissipative(resp., dissipative)with respect to the supply rates(u, y)=qTy+rTu if and only if there existp ∈ R

n

+, l ∈ Rn

+, andw ∈ Rm

+, and a scalarε >0 (resp.,ε = 0) such that

0= ATp + εp − CTq + l, (39)

0= BTp − DTq − r + w. (40)

Proof. Sufficiency follows from Theorem 5.1 withf (x) = Ax, G(x) = B, h(x) = Cx,J (x)=D, andVs(x)=pTx. To shownecessity, note that if the linear nonnegative dynamicalsystem (37), (38) is dissipative with respect to the linear supply rates(u, y) = qTy + rTu,then it follows from Theorem 5.3 withf (x) = Ax,G(x) = B, h(x) = Cx, andJ (x) = D

that there existp ∈ Rn

+, l ∈ Rn

+, andw ∈ Rm

+ such that (39) and (40) are satisfied.�

Remark 6.1. For a givenl ∈ Rn andw ∈ Rm, note that there existsp ∈ Rn such that (39),(40) are satisfied if and only if rank[M y] = rankM, where

M�=

[(A + εIn)

T

BT

], y

�=[

CTq − l

DTq + r − w

].

Now, there existp� �0, l� �0, andw� �0 such that (39), (40) are satisfied if and onlyif the inequalities

p� �0, (41)

z − Mp� �0, (42)

where

z�=

[CTq

DTq + r

],

are satisfied. Eqs. (41) and (42) comprise a set of 2n + m linear inequalities withpi ,i =1, . . . , n, variables and hence the feasibility ofp� �0 such that (41) and (42) hold canbe checked by standard linear matrix inequality (LMI) techniques[10].


Remark 6.2. An identical theorem to Theorem 6.2 holds for lossless systems with linearsupply ratess(u, y) = qTy + rTu. In this case, (39) and (40) hold withl = 0 andw = 0.

7. Feedback interconnections of nonnegative dynamical systems

Feedback systems are pervasive in nature and can be found almost everywhere in livingsystems. In particular, control at the intercellular level, DNA replication and cell division,control of gene expression, control of enzyme activity, control at the organ system andorganism level, humoral control, neural control, and regulation in biological systems allinvolve feedback systems[28]. To analyze these complex nonnegative systems, the notionof dissipativity, with appropriate storage functions and supply rates, can be used to constructLyapunov functions by appropriately combining storage functions for each subsystem andexamining their respective structure. In this section we consider stability of feedback inter-connections of nonnegative dynamical systems. We begin by considering the nonnegativedynamical systemG given by (16), (17) with the nonlinear nonnegative dynamical feedbacksystemGc given by

xc(t) = fc(xc(t)) + Gc(xc(t))uc(t), xc(0) = xc0, t�0, (43)

yc(t) = hc(xc(t)), (44)

wherefc : Rnc → Rnc, Gc : Rnc → Rnc×mc, hc : Rnc → Rlc, fc is essentially nonnega-tive,Gc(xc)� �0, andhc(xc)� �0, xc ∈ R

nc+ .

Theorem 7.1. Let q ∈ Rl , r ∈ Rm, qc ∈ Rlc, and rc ∈ Rmc. Consider the nonlinearnonnegative dynamical systemsG andGc given by(16), (17),and(43), (44),respectively.Assume thatG is dissipative with respect to the linear supply rates(u, y) = qTy + rTu

and with a positive-definite storage functionVs(·), and assume thatGc is dissipative withrespect to the linear supply ratesc(uc, yc)=qTc yc+rTc uc andwith a positive-definite storagefunctionVsc(·). Then the following statements hold:(i) If there exists a scalar�>0such thatq+�rc� �0andr+�qc� �0, then the positive

feedback interconnection ofG andGc is Lyapunov stable.(ii) If G andGc are zero-state observable and there exists a scalar�>0 such thatq +

�rc>0 and r + �qc>0, then the positive feedback interconnection ofG andGc isasymptotically stable.

(iii) If G is zero-state observable, rankGc(0) = mc, Gc is exponentially dissipative withrespect to the supply ratesc(uc, yc) = qTc yc + rTc uc, and there exists a scalar�>0such thatq + �rc� �0 andr + �qc� �0, then the positive feedback interconnectionofG andGc is asymptotically stable.

(iv) If G is exponentially dissipative with respect to the supply rates(u, y)=qTy+rTu,Gcis exponentially dissipative with respect to the supply ratesc(uc, yc) = qTc yc + rTc uc,and there exists a scalar�>0 such thatq + �rc� �0 and r + �qc� �0, then thepositive feedback interconnection ofG andGc is asymptotically stable.


Proof. Note that the positive feedback interconnection ofG andGc is given byu= yc anduc = y so that the closed-loop dynamics ofG andGc is given by[

x(t)

xc(t)

]=

[f (x(t)) + G(x(t))hc(xc(t))

fc(xc(t)) + Gc(xc(t))h(x(t)) + Gc(xc(t))J (x(t))hc(xc(t))

]�= f (x(t)),

wherex =[xT xTc ]T, which implies thatf (x) is essentially nonnegative. Hence, the closed-loop system is also nonnegative and thusx(t)� �0, xc(t)� �0, u(t)� �0, y(t)� �0,t�0. Now, the proof follows from Lyapunov theory and invariant set theorem argumentsusing the Lyapunov function candidateV (x, xc) = Vs(x) + �Vsc(xc). �

The following corollary to Theorem 7.1 addresses linear supply rates of the forms(u, y) = eTu − eTy.

Corollary 7.1. Consider the nonlinear nonnegative dynamical systemsG andGc given by(16), (17),and(43), (46),respectively. Assume thatG is nonaccumulative with a positive-definite storage functionVs(·), and assume thatGc is exponentially nonaccumulative witha positive-definite storage functionVsc(·). Then the following statements hold:(i) If G is zero-state observable and rankGc(0)=mc, then the positive feedback intercon-

nection ofG andGc is asymptotically stable.(ii) If G is exponentially nonaccumulative, then the positive feedback interconnection ofG

andGc is asymptotically stable.

Proof. The proof is a direct consequence of (iii) and (iv) of Theorem 7.1 with� = 1,q = −rc = −e, andr = −qc = e. �

Next, we develop absolute stability criteria for nonnegative feedback systems with non-negative time-varying memoryless input nonlinearities. Since absolute stability theory con-cerns the stability for classes of feedback nonlinearities which, as noted in[23], can readilybe interpreted as an uncertainty model, the proposed framework can be used to analyzerobustness of biological and physiological systems developed from data models. Specifi-cally, given the linear nonnegative systemG characterized by (37), (38) we derive sufficientconditions that guarantee asymptotic stability of the feedback interconnection involving thelinear nonnegative systemG and the feedback nonnegative time-varying input nonlinearity�(·, ·) ∈ �, where

��= {� : R+ × R

l

+ → Rm

+: �(·,0) = 0, 0� ��(t, y)� �My, y ∈ Rl

+,

a.e.t�0, and�(·, y) is Lebesgue measurable for ally ∈ Rl

+}, (45)

M?0, andM ∈ Rm×l .

Theorem 7.2. Consider the nonnegative dynamical systemG given by(37), (38)and as-sume that(A,C) is observable andG is exponentially dissipative with respect to the supplyrate s(u, y) = eTu − eTMy, whereM?0.Then, the positive feedback interconnection ofG and�(·, ·) is globally asymptotically stable for all�(·, ·) ∈ �.


Proof. Since�(t, y)� �0 for all t�0,y ∈ Rl

+, and (37), (38) is a nonnegative dynamicalsystem, it follows that the positive feedback interconnection ofG and�(·, ·) given by

x(t) = Ax(t) + B�(t, y(t)), x(0) = x0, t�0,

is a nonnegative dynamical system for all�(·, ·) ∈ �. Next, since(A,C) is observableandG is exponentially dissipative with respect to the supply rates(u, y) = eTu − eTMy,it follows from Remark 5.3 and Theorem 6.2, withr = e andq = −MTe, that there existsp ∈ Rn+, l ∈ R

n

+, andw ∈ Rm

+, and a scalarε >0 such that

0= ATp + εp + CTMTe + l, (46)

0= BTp + DTMTe − e + w. (47)

Next, consider the Lyapunov function candidateVs(x) = pTx and note that the Lyapunovderivative satisfies

Vs(x) = pT(Ax + B�)= − εpTx + eT[� − My] − lTx − wT�� − εVs(x) + eT[� − My]. (48)

Now, sinceVs(x)>0, x ∈ Rn

+, x = 0, and�� My for all �(·, ·) ∈ �, it follows thatVs(x)<0,x ∈ R

n

+, x = 0. Hence, the positive feedback interconnection ofG and�(·, ·) isglobally asymptotically stable for all�(·, ·) ∈ �. �

Remark 7.1. To consider nonlinearities with upper and lower bounds of the formM1y

� ��(t, y)� �M2y, where�(·, ·) ∈ �, we can use the standard loop shifting techniquesdiscussed in[33, p. 408]. In this case, Theorem 7.2 holds with�(t, y), A, B, C,D, andMreplaced by�(t, y) − M1y, A + B(I − M1D)−1M1C, B(I − M1D)−1, (I − DM1)

−1C,(I − DM1)

−1D, andM2 − M1, respectively.

Theorem 7.3. Consider the nonlinear nonnegative dynamical systemG given by(16), (17)and assume thatG is zero-state observable and exponentially dissipative with respect to thesupply rates(u, y)=eTu−eTMy,whereM?0.Then, the positive feedback interconnectionofG and�(·, ·) is globally asymptotically stable for all�(·, ·) ∈ �.

Proof. The proof is similar to that of Theorem 7.2 and hence is omitted.�

8. Illustrative examples

In this section, we provide two examples that demonstrate the utility of the basic mathe-matical results developed in the paper.

Example 8.1. Consider the general formof a two-compartmentmodelwith arbitrary inputsandoutflows (elimination) shown inFig. 1,where fori, j=1,2,xi is the sizeof compartmenti in mass units,aij is the instantaneous transfer coefficient of material flow fromj to i inunits of (time)−1,aii is the flow loss coefficient fromcompartmenti out of the system inunits


of (time)−1, andui is the instantaneous rate of flow of material from outside the system intothe compartmenti in units of mass/time. A mass balance for the two-compartment systemyields (16) withx = [x1 x2]T, u = [u1 u2]T, and

A =[−(a11+ a21) a12

a21 −(a22+ a12)

], B =

[1 00 1

].

This model is frequently used to analyze the distribution or flow of a drug or other material(the tracer) through thehumanbodyafter injection into thebloodstream[28].Theflowof thematerial to be studied is assumed to be near steady state; that is, the material is mixed in theblood plasma so that the amount in the plasma is at a uniform concentration. Furthermore,the tracer is introduced in a comparatively small quantity so that even if the original flows arenonlinear, the tracer flows can be modeled by linear equations. A simple example of such amodel would be of a lipoproteinmetabolism. In this case, the first compartment correspondsto the blood plasma and the second compartment corresponds to the extravascular space.Here we assume that the tracer is introduced intravenously as an impulsive injection; thatis, a bolus injection, into the blood stream and henceu2 = 0. Furthermore, we assume thatthe tracer does not enter cells so thata22 = 0. Finally, we assume that the compound isremoved from the plasma by excretion into the kidneys and the amount excreted is somefraction of the amount filtered in the renal glomeruli so thata11>0.To analyze this system, first note thatA is nonsingular and essentially nonnegative.

Furthermore, since the input material is a bolus injection we can always reproduce theimpulsive response with the free response by settingx(0)=Bv, wherev ∈ Rm denotes theimpulse strength.Hence, theabovemodel canbeanalyzedasan input-closed compartmentalmodel. Now, it follows from Lemma 2.2 that eAt � �0 for all t�0 and consequently ifx(0)is nonnegative, then the solutionx(t) = eAtx(0) is nonnegative for allt�0. Furthermore,sinceA is a nonsingularmatrix it follows that the set of equilibriaE={(x1, x2) ∈ R

2+: Ax=

0} = N(A) = {(0,0)}. Next, takingp = e?0 andr = [a11 0]T� �0, it follows that (2)holds and hence, since(A, rT) is observable, the system is asymptotically stable by (iv)of Theorem 3.2. Finally, we show that the system is nonaccumulative. However, since thematerial outflowof thefirst compartment corresponding tobloodplasmacannot bemeasureddirectly, wemeasure the concentration of the material; that is,y=a11x1/v1, wherev1 is thevolume of distribution of the blood plasma in the first compartment. Now, using the storagefunctionVs(x1, x2) = x1 + x2 it follows that the lipoprotein metabolism model is losslesswith respect to the supply rates(u, y) = u1 − v1y.Finally, we use the two-compartment model shown inFig. 1 to study the behavior of a

closed compartmental system; that is,u1=u2=0 anda11=a22=0. These systems can arisewhen studying potassium ions continually moving from the plasma into the red blood cellsand vice versa in the human bloodstream. Since potassium is concentrated in red blood cellsby a nonlinear active transport mechanism, the resulting compartmental model is inherentlynonlinear. However, experimental data reported in[45] show that potassium levels in boththe plasma and red cells stay relatively constant over time and hence at steady state the tracerdistribution is linear. Hence, we use the model inFig. 1with u1=u2=0 anda11= a22=0to analyze this system with the total red blood cells containing an amountx1 of potassium(the tracee) designated as the first compartment, and blood plasma, containing a quantity


x2 of potassium, as the second compartment. In this case,A is singular and essentiallynonnegative. Hence,x(t)= eAtx(0)= [I2+ (1− e−(a21+a12)t /a21+ a12)A]x0� �0 for allt�0. Furthermore, the set of equilibria are givenbyE={(x1, x2) ∈ R

2+: Ax=0}=N(A)=

{(x1, x2) ∈ R2+: x2=a21/a12x1}. Finally, takingp=e?0 andr=[0 0]T, it follows that (2)

holds and hence the system is semistable by (ii) of Theorem 3.2.Alternatively, semistabilityalso follows by noting that limt→∞ eAt = I2 + (1/(a21+ a12))A exists.

Example 8.2. This example considers a nonlinear compartmentalmodel for epidemiology;that is, the spread of epidemics. There are several such models in the literature and herewe consider the basic Susceptible–Infected–Removed (SIR) model[28]. Specifically, themodel of an SIR epidemic is given by

x1(t) = − �N

x1(t)x2(t) − x1(t) + u(t), x1(0) = x01, t�0, (49)

x2(t) = �N

x1(t)x2(t) − ( + )x2(t), x2(0) = x02, (50)

x3(t) = x2(t) − x3(t), x3(0) = x03, (51)

wherex1 denotes thenumber of susceptibles,x2 denotes thenumber of infectives,x3 denotesthe number of immunes, >0 is the death rate coefficient,N is a constant denoting thetotal size of the population; that is,x1(t) + x2(t) + x3(t) = N , u is the rate of recruitmentof new members into the susceptible pool and is assumed to be a constant rate that justmakes up for the deaths; that is,u(t) = u = N , >0 is a rate constant for recovery, and�>0 is the mean contact rate per person for contacts that transmit the disease. Note thatthe nonlinear terms in (49) and (50) arise due to the fact that the rate of susceptibles thatbecome infected is�x1x2/N ; that is, the product of the rate of infected that make contactand transmit the disease(�x2) and the fraction of susceptibles in the population(x1/N).To analyze this system, first note that the vector field of (49)–(51) is essentially nonneg-

ative. Furthermore, sincex1(t) + x2(t) + x3(t) = N , (51) is superfluous and we need onlyconsider (49) and (50). In addition note that (50) can be equivalently written as

x2(t) = ( + )[ �N

x1(t) − 1]x2(t), x2(0) = x02, (52)

where��= �/(+ ) represents the number of infections that are transmitted by an infective,

over the lifetime of the infection, if all contacts are with susceptibles. Now, the set of

equilibria for (49) and (50) areE = {(x1, x2) ∈ R2+: f (x) = 0} = {(N,0)} if ��1 and

E = {(N,0), (N/�, N(� − 1)/�)} if �>1.In the case where�>1, linearizing (49), (50) about the two equilibria and computing the

eigenvalues of the resulting Jacobian matrix, it follows from Theorem 4.1 that the equilib-riumpoint(N,0)of theSIRmodel is unstableand theequilibriumpoint(N/�, N(�−1)/�)is locally asymptotically stable. Hence, in this case, there exists a stable epidemic level.Next,weconsider thecasewhere��1.Consider theLyapunov functioncandidateV (x)=

12(x1−N)2+Nx2 andnote that since��1 it follows thatV (x)=− (x1−N)2−�/Nx2(x1−N)2 + N( + )(� − 1)x2�0, x ∈ R

2+, which shows that the equilibrium point(N,0) is


Lyapunov stable. Now, consider the functionE(x)=x2 and note that sincex1�N it followsthat E(x) = ( + )[�/Nx1 − 1]x2�0. Next, using LaSalle’s invariance principle[33] itcan be shown that the largest invariant setM contained in the setR

�={x ∈ R2+: E(x)=0}

is given byM = {(N,0)} which shows that(N,0) is a globally asymptotically stableequilibrium. Hence, in the case where��1, a disease introduced into the population diesout; that is, it cannot propagate and gives rise to an epidemic.Next, we consider the case corresponding to zero death rate; that is,u= =0. In this case,

the system equilibria are given byE= {(x1, x2) ∈ R2+: x2= 0}. First, using the Lyapunov

function candidateV (x)=1/2(x1−x1e)2+N/�x2, it can be shown that every point in the

setEs�={(x1e, x2e) ∈ R

2+: x1e�N/�, x2e= 0} is Lyapunov stable. Alternatively, using

Theorem6.1 it can be shown that every point in the set{(x1, x2) ∈ R2+: x1> N/�, x2=0}

is unstable. To further analyze these equilibria, consider the functionE(x) = x1 + x2 and

note thatE(x) = −x2�0, x ∈ R2+. Now, applying LaSalle’s invariance principle[33]it

can be shown that the largest invariant setM contained in the setR�={x ∈ R

2+: E(x)=0}

is given byM=E. Furthermore, sincex1(t) is bounded and monotonic it follows from theBolzano-Weierstrass theorem that limt→∞x1(t) exists. Hence, for every initial condition

x(0) ∈ R2+, limt→∞x(t) exists, limt→∞x2(t) = 0, and limt→∞x1(t) + x3(t) = N . In

addition, it can be shown that limt→∞x1(t)�N/� which shows that every equilibriumpoint inEs is semistable.Finally, we show that the SIRmodel is nonaccumulative foru(t) ∈ R, t�0. Specifically,

withmeasured outputsy= x2 and storage functionVs(x1, x2, x3)=x1+x2+x3, it followsthatVs(x1, x2, x3) = u − y − x1 − x3 which implies thatVs(x1, x2, x3)�u − y.

9. Conclusion

Nonnegative dynamical systemsare used to capture the dynamics of systemswhose statesarenonnegative.Thesesystemsarewidespread inbiologyandmedicineandarederived frommassandenergybalanceconsiderations involving theexchangeof nonnegativequantities. Inthis paper we developed stability results for linear and nonlinear nonnegative systems usinglinear Lyapunov functions. In addition, necessary and sufficient conditions for dissipativitywith linear and nonlinear storage functions and linear supply rates were also developed.Finally, general stability criteria were given for Lyapunov, semi, and asymptotic stabilityof feedback interconnections of linear and nonlinear nonnegative dynamical systems.

Notation

R,Rm×n set of real numbers, set ofm × n real matricesxi, e ith entry of vectorx ∈ Rn, [1, 1, . . . ,1]Tei vector with unity inith position and zeros elsewhereA(i,j) (i, j)th entry of matrixA ∈ Rm×n

A� �0 (A?0) A(i,j)�0 (A(i,j) >0) for all i andj


A� �B (A?B) A − B� �0 (A − B?0) whereA andB are matriceswith identical dimensions

A�0 (A>0) nonnegative (resp., positive) definite matrix; that is, symmetricmatrix with nonnegative (resp., positive) eigenvalues

A�B (A>B) A − B�0 (A − B >0) whereA andB are symmetric matriceswith identical dimensions

Rn+, Rn

+, {x ∈ Rn: x?0}, {x ∈ Rn: x� �0}spec(A) spectrum of matrixA ∈ Rn×n

�(A) spectral radius ofA; that is, max{|�| : � ∈ spec(A)}rankA, ind(A) rank of matrixA ∈ Rm×n, min{k ∈ N: rankAk = rankAk+1}det(A) determinant of matrixA ∈ Rn×n

rowi (A), coli (A) ith row ofA, ith column ofAAT, A# transpose ofA, group generalized inverse ofA where ind(A)�1R(A),N(A) range and null subspaces ofA ∈ Rm×n

A ⊕ B Kronecker sum ofA andB‖ · ‖ vector or matrix normBr (x0),S {x ∈ Rn: ‖x − x0‖<r}, closure of the setS

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