Bisimulation Equivalence of Discrete-Time Stochastic Linear Control Systems

In this paper, we propose a definition of equivalence via stochastic bisimulation for the class of discrete-time stochastic linear control systems with possibly degenerate normally distributed disturbances. The notion is inspired by the notion of probabilistic bisimulation for probabilistic chains. Geometric necessary and sufficient conditions for checking this notion are derived. Model reduction via Kalman-like decomposition is also proposed. Connections with stochastic reachability are discussed and with finite horizon stochastic safety problems established. A discussion on the use of stochastic bisimulation equivalence for control design is given, and an application to optimal control problems with stochastic reachability specifications is finally presented.