Stabilizing Rate of Stochastic Control Systems

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.

💡 Research Summary

This paper presents a comprehensive quantitative framework for analyzing the optimal stabilizing rate in stochastic linear systems with multiplicative noise. The optimal stabilizing rate is defined as the fastest achievable mean-square exponential decay rate of the system state under admissible control policies, moving beyond the traditional binary question of stabilizability to address “how fast” stabilization can be achieved.

The core contributions are structured in a two-pronged approach. First, the authors extend norm-based techniques from the analysis of deterministic switched systems to the stochastic setting. This extension is nontrivial because the natural state space for the second moment is infinite-dimensional (L2), where norms are not equivalent, unlike in finite-dimensional spaces. They establish conditions under which a form of norm equivalence can be recovered and derive a verifiable necessary and sufficient condition for the exact attainability of the optimal rate, along with computable upper and lower bounds.

Second, focusing on state-feedback control policies, the problem is reformulated as an optimal control problem with a nonlinear cost functional, leading to a Bellman-type equation. This equation is not directly tractable but can be recast as a nonlinear matrix eigenvalue problem that requires a strictly positive-definite matrix solution. A significant theoretical hurdle arises here: such a positive-definite solution may not exist for the original problem, leading to ill-posedness on the boundary of the positive semidefinite cone.

To overcome this, the paper introduces a regularization scheme. By adding a small perturbation parameter (ε > 0), the modified problem is guaranteed to admit a strictly positive-definite solution. To compute this solution constructively, the authors develop the Regularized Normalized Value Iteration (RNVI) algorithm. RNVI operates via a continuation method: it starts with a relatively large regularization parameter where convergence is numerically robust, computes the fixed point (a positive-definite matrix) and the corresponding feedback controller, and then uses this solution to initialize the computation for a slightly smaller ε. This process continues, generating a sequence of controllers and associated matrix solutions.

A key strength of the framework is that each regularized solution, obtained for a specific ε, yields certified upper and lower bounds for the true optimal stabilizing rate ρ*. As ε decreases, these bounds become tighter. Therefore, the RNVI algorithm not only produces feasible feedback controllers but also provides a verifiable interval that contains the optimal performance limit ρ*. This creates a constructive and verifiable methodology for quantifying the fastest achievable mean-square stabilization under multiplicative noise.

The paper concludes with numerical experiments demonstrating the effectiveness of the RNVI algorithm in generating increasingly precise bounds for the optimal stabilizing rate, thereby validating the proposed theoretical framework and computational tool.