Stability Analysis of Stochastic Optimal Control: The Linear Discounted Quadratic Case

In this article, we analyze the stability properties of stochastic linear systems in closed loop with an optimal policy that minimizes a discounted quadratic cost in expectation. In particular, the linear system is perturbed by both additive and multiplicative stochastic disturbances. We provide conditions under which mean-square boundedness, mean-square stability, and recurrence properties hold for the closed-loop system. We distinguish two cases, when these properties are verified for any value of the discount factor sufficiently close to 1, or when they hold for a fixed value of the discount factor in which case tighter conditions are derived, as illustrated in an example. The analysis exploits properties of the optimal value function, as well as a detectability property of the system with respect to the stage cost, to construct a Lyapunov function for the stochastic linear quadratic regulator problem.