1 Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay Wassim M. Haddad † and VijaySekhar Chellaboina * † School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150 * Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211-2200 Abstract— Nonnegative and compartmental dynamical sys- tem models are derived from mass and energy balance considerations and involve the exchange of nonnegative quan- tities between subsystems or compartments. These models are widespread in biological and physical sciences and play a key role in understanding these processes. A key phys- ical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. In this paper we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay. Specifically, asymptotic stability conditions for linear and nonlinear as well as continuous-time and discrete-time nonnegative dynamical systems with time delay are established using linear Lyapunov-Krasovskii functionals. I. I NTRODUCTION Nonnegative and compartmental models play a key role in understanding many processes in biological and medical sciences [1]–[6]. Such models are composed of homoge- neous interconnected subsystems (or compartments) which exchange variable nonnegative quantities of material with conservation laws describing transfer, accumulation, and outflows between compartments and the environment. The range of applications of nonnegative and compartmental systems is not limited to biological and medical systems. Their usage includes chemical reaction systems, queuing systems, ecological systems, economic systems, telecom- munication systems, transportation systems, and power sys- tems, to cite but a few examples. A key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dy- namics of such systems should account for material, energy, or information in transit between compartments [3]. Hence, to accurately describe the evolution of the aforementioned systems, it is necessary to include in any mathematical model of the system dynamics some information of the past system states. This of course leads to (infinite-dimensional) delay dynamical systems [7]–[9]. In this paper we develop necessary and sufficient condi- tions for time-delay nonnegative and compartmental dynam- ical systems. Specifically, using linear Lyapunov-Krasovskii functionals we develop necessary and sufficient conditions for asymptotic stability of linear nonnegative dynamical systems with time delay. The consideration of a linear This research was supported in part by NSF under Grants ECS-9496249 and ECS-0133038 and AFOSR under Grant F49620-0-01-0095. Lyapunov-Krasovskii functional leads to a new Lyapunov- like equation for examining stability of time delay non- negative dynamical systems. The motivation for using a linear Lyapunov-Krasovskii functional follows from the fact that the (infinite-dimensional) state of a retarded non- negative dynamical system is nonnegative and hence a linear Lyapunov-Krasovskii functional is a valid candidate Lyapunov-Krasovskii functional. For a time delay com- partmental system, a linear Lyapunov-Krasovskii functional is shown to correspond to the total mass of the system at a given time plus the integral of the mass flow in transit between compartments over the time intervals it takes for the mass to flow through the intercompartmental connections. The contents of the paper are as follows. In Section II we establish definitions, notation, and review some basic results on nonnegative dynamical systems. In Section III we show that for a nonnegative continuous function specifying the initial state of a retarded nonnegative system, time delay nonnegative and compartmental systems are confined to a nonnegative state space. Furthermore, we give necessary and sufficient conditions for asymptotic stability for linear time delay nonnegative systems using a linear Lyapunov- Krasovskii functional and a new Lyapunov-like equation. We then turn our attention to nonlinear nonnegative sys- tems with time delay and present sufficient conditions for asymptotic stability. In Section IV we present a discrete- time analog of the results developed in Section III. Finally, we draw conclusions in Section V. II. MATHEMATICAL PRELIMINARIES In this section we introduce notation, several definitions, and some key results concerning linear nonnegative dy- namical systems [1], [5], [6], [10] that are necessary for developing the main results of this paper. Specifically, N denotes the set of nonnegative integers, R denotes the reals, and R n is an n-dimensional linear vector space over the reals with the maximum modulus norm k·k given by kxk = max i=1,...,n |x i |, x ∈ R n . For x ∈ R n 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 nonnegative or positive, respectively. Likewise, A ∈ R n×m is nonnegative or positive if every entry of A is nonnegative or positive, respectively, which is written as A ≥≥ 0 or A >> 0, respectively. Let R n + and R n + denote the nonnegative and positive orthants of R n ; that is, if x ∈ R n , then x ∈ R n + and x ∈ R n + are equivalent, respectively, to x ≥≥ 0 and x >> 0. Finally, C ([a, b], R n ) denotes a Banach space of continuous functions mapping the interval [a, b] into R n with the topology of uniform convergence.