Stabilizing Rate of Stochastic Control Systems

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. The framework consists of two complementary developments. First, we extend the norm-based analysis from deterministic switched systems to the stochastic setting and establish computable upper and lower bounds for the optimal stabilizing rate. Second, by restricting attention to state-feedback policies, we introduce an optimal control formulation of the optimal stabilizing rate problem 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 overcome the possible absence 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 the 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, providing a constructive procedure for estimating the fastest achievable mean-square decay rate. We also provide a sufficient condition for the certified gap to close and a necessary structural condition satisfied by each regular nonvanishing-gap fixed-point sequence. Numerical experiments further demonstrate the effectiveness of the proposed framework.

47 pages