Stability and dissipativity theory for nonnegative dynamical systems: a thermodynamic framework for biological and physiological systems

Abstract

Nonnegative dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models 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 interconnections of linear and nonlinear nonnegative dynamical systems. Specifically, using linear Lyapunov functions we develop necessary and sufficient conditions for Lyapunov stability, semistability, and asymptotic stability for nonnegative systems. In addition, using linear storage functions and linear supply rates we develop, new notions of dissipativity theory for nonnegative dynamical systems. Finally, these results are used to develop general stability criteria for feedback.