Stabilizing Rate of Stochastic Control Systems

TL;DR

The paper tackles the problem of fastest mean-square exponential stabilization for discrete-time stochastic systems with multiplicative noise by introducing the optimal stabilizing rate ρ*. It develops a dual framework: (i) a norm-based extension of deterministic results to the stochastic setting, providing attainability criteria and computable bounds, and (ii) an optimal-control formulation with a Bellman-type equation that reduces to a nonlinear matrix eigenvalue problem solvable with Regularized Normalized Value Iteration (RNVI). The RNVI approach yields strictly positive-definite fixed points and certified upper and lower bounds for ρ*, validated through 2D and 4D numerical experiments near and across stability boundaries. Collectively, the work delivers a constructive, verifiable toolkit for determining and achieving the fastest possible mean-square stabilization under multiplicative disturbances, with practical gains and exponential convergence demonstrated in simulations.

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.

Figures (10)

• Figure 1: Bounds for $\rho^*$ over the $\tau$-grid for different noise levels $\sigma$.
• Figure 2: Certified squared-gap $\Delta_\tau=\frac{\tau}{n(1-\tau)}\lambda_{\max}((P^{(\tau)})^{-1})$ for each noise level $\sigma$.
• Figure 3: $\lambda_{\max}((P^{(\tau)})^{-1})=1/\lambda_{\min}(P^{(\tau)})$ for each noise level $\sigma$.
• Figure 4: RNVI iteration counts for $\sigma=3$ across the $\tau$–grid.
• Figure 5: Mean-square energy trajectory in logarithmic scale ($\sigma = 2$).

Theorems & Definitions (52)

• Definition 1
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• Lemma 1