Lyapunov Theorems for Semistability of Discrete-Time Stochastic Systems with Application to Network Consensus with Random Communication Noise

This paper develops Lyapunov and converse Lyapunov theorems for discrete-time stochastic semistable nonlinear dynamical systems expressed by Itô-type difference equations possessing a continuum of equilibria. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose difference operator involves a discrete-time analog of the infinitesimal generator for continuous-time Ito dynamical systems and decreases along the dynamical system sample solution sequences satisfying an inequality involving the average distance to the set of the system equilibria. These results are then used to develop semistable consensus protocols for discrete-time networks with communication uncertainty capturing measurement noise and attenuation errors in the information transfer between agents. The proposed distributed control architecture involves the exchange of information between agents guaranteeing that the closed-loop dynamical network is stochastically semistable to an equipartitioned equilibrium representing a state of almost sure consensus consistent with basic discrete-time thermodynamic principles.