Dissipativity theory for discrete-time nonlinear stochastic dynamical systems

In this paper, we develop stochastic dissipativity theory for discrete-time nonlinear dynamical systems expressed by Ito-type difference equations using basic state properties. Specifically, a stochastic version of dissipativity and losslessness using a state dissipation inequality in expectation for controlled Markov chains is presented. The results are then used to derive extended Kalman-Yakubovich-Popov conditions for characterizing stochastic dissipativity and losslessness in terms of the nonlinear system functions using continuous storage functions. Finally, feedback interconnection stability in probability results are developed generalizing the small gain and positivity theorems to discrete-time stochastic systems.