Stabilizing Rate of Stochastic Control Systems with Multiplicative Noise

Abstract: This paper develops a quantitative framework for analyzing the mean-square exponential stabilization of stochastic linear systems with multiplicative noise, focusing specifically on the optimal stabilizing rate, which characterizes the fastest exponential stabilization achievable under admissible control policies. Our contributions are twofold. First, we extend norm-based techniques from deterministic switched systems to the stochastic setting, deriving a verifiable necessary and sufficient condition for the exact attainability of the optimal stabilizing rate, together with computable upper and lower bounds. Second, by restricting attention to state-feedback policies, we reformulate the optimal stabilizing rate problem as an optimal control problem with a nonlinear cost function and derive a Bellman-type equation. Since this Bellman-type equation is not directly tractable, we recast it as a nonlinear matrix eigenvalue problem whose valid solutions require strictly positive-definite matrices. To ensure the existence of such solutions, we introduce a regularization scheme and develop a Regularized Normalized Value Iteration (RNVI) algorithm, which in turn generates strictly positive-definite fixed points for a perturbed version of original nonlinear matrix eigenvalue problem while producing feedback controllers. Evaluating these regularized solutions further yields certified lower and upper bounds for the optimal stabilizing rate, resulting in a constructive and verifiable framework for determining the fastest achievable mean-square stabilization under multiplicative noise.