Reachability and Controllability Analysis of the State Covariance for Linear Stochastic Systems

Abstract: This paper studies the set of terminal state covariances that are reachable over a finite time horizon from a given initial state covariance for a linear stochastic system with additive noise. For discrete-time systems, a complete characterization of the set of reachable state covariances is given. For continuous-time systems, we present an upper bound on the set of reachable state covariances. Moreover, for both linear discrete-time and continuous-time systems, necessary and sufficient conditions are provided for the controllability of the state covariance over a finite horizon.