Dissipative Stochastic Dynamical Systems

Dissipative dynamical systems provide fundamental connections between physics, dynamical systems theory, and control science and engineering. In the deterministic setting, dissipativity theory has been extensively developed in the literature to provide a general framework for the analysis and design of control systems using an input-state-output system description based on generalized system energy considerations that uses a state-space formalism to link engineering systems with memory to well known physical phenomena. Recently, several results have appeared in the literature extending dissipativity notions to the stochastic setting in order to develop an analogous theory for stochastic dynamical systems. However, unlike the deterministic theory, which can involve either an energy balance or a power balance state dissipation inequality for characterizing system dissipativity, in the stochastic case this equivalence is far more nuanced. In this paper, we develop a general theory for stochastic dissipativity and present general conditions on the system drift and diffusion functions as well as the system energy storage and supply rates to provide an equivalence between the sample path dependent energetic (i.e., supermartingale) and the power balance (i.e., algebraic) forms for characterizing stochastic dissipativity.