Dissipativity Theory for Discrete-Time Nonlinear Stochastic Dynamical Systems, Part I: Input-Output and State Properties

In this paper, we develop stochastic dissipativity theory for discrete-time nonlinear dynamical systems expressed by Itô-type difference equations using basic input-output and state properties. Specifically, a stochastic version of dissipativity and losslessness using both an input-output as well as a state dissipation inequality in expectation for controlled Markov chains is presented. In particular, using generalized notions of average system energy storage and average external energy supply, we present a systematic treatment of stochastic dissipativity theory as a generalization of classical deterministic dissipativity theory. These results are then used in the second part of this paper to develop feedback interconnection stability in probability results as well as extensions of the small gain and positivity theorems for discrete-time stochastic systems.