Dissipativity Theory for Nonlinear Stochastic Dynamical Systems

In this paper, we develop stochastic dissipativity theory for nonlinear dynamical systems using basic input-output and state properties. Specifically, a stochastic version of dissipativity using both an input-output as well as a state dissipation inequality in expectation for controlled Markov diffusion processes is presented. The results are then used to derive extended Kalman-Yakubovich-Popov conditions for characterizing necessary and sufficient conditions for stochastic dissipativity of stochastic dynamical systems using two-times continuously differentiable storage functions. In addition, feedback interconnection stability in probability results for stochastic dynamical systems are developed thereby providing a generalization of the small gain and positivity theorems to stochastic systems.