An Inner-Outer Iteration Algorithm with Optimal Parameters for Stochastic Lyapunov Matrix Equation

Jul 16, 2026

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This paper proposes an inner–outer (IO) iterative algorithm with optimal parameters for solving stochastic Lyapunov matrix equations associated with discrete-time stochastic linear systems. Assuming asymptotic mean-square stability, the work analyzes monotonicity and boundedness of the generated iterates and proves a sufficient convergence result for zero initial conditions. By using the spectral radius of the induced iteration matrix, it derives necessary and sufficient convergence criteria for arbitrary initial conditions. Parameter-selection strategies and numerical examples confirm improved convergence versus existing methods.

LanguageEnglish

CategoryResearch & Report

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Pages8

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Introduction

Previous results and the proposed algorithm

An Inner-Outer Iteration Algorithm with Optimal Parameters for Stochastic Lyapunov Matrix Equation⋆ Donghuan He , Xiaowen Su , Feng Wang and Xuesong Chen∗ arXiv :2607 .00451v1 [math .NA] 1 Jul 2026 School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, P.R. China | ARTICLE INFO | | AB STRACT | | --- | --- | --- | | Keywords:<br>Linear/nonlinear models N-dimensional systems Stochastic control Iterative schemes Parametric optimization | | This paper proposes an inner–outer (IO) iterative algorithm with optimal parameters for solving stochastic Lyapunov matrix equation associated with discrete-time stochastic linear system. First, under the assumption that the underlying stochastic linear system is asymptotically mean-square stable, the monotonicity and boundedness of the iterative sequence generated by the proposed algorithm are analyzed. On this basis, a sufficient convergence result is established for the zero initial condition. Second, by deriving the spectral radius of the corresponding iteration matrix, several necessary and sufficient convergence conditions are obtained for arbitrary initial conditions. In addition, the optimal parameter-selection strategies are developed to improve the convergence performance of the algorithm. Finally, numerical examples are presented to verify the theoretical results and demonstrate the advantages of the proposed algorithm over several existing iterative methods. | 1. Introduction Stochastic linear systems have been widely applied in various fields.For instance, they have been used to construct dynamic segment models for speech recognition Digalakiset al. (2002) and applied to real-time control with deadline overruns Gallant et al. (2025) . Owing to their importance, the stability, observability, and detectability of stochastic linear systems have attracted considerable attention. McLane (1969) introduced the concept of asymptotic mean-square stability (AMSS) for linear stochastic systems. It was further proved that the existence of a positive definite solution to the corresponding Lyapunov matrix (LM) equation provides a necessary and sufficient condition for the AMSS of the system. These results indicate that the properties of stochastic linear systems are closely related to those of the solutions to their corresponding LM equations. Such equations are referred to as stochastic Lyapunov matrix (SLM) equations. Therefore, developing efficient methods for solving SLM equations is of great importance. A direct method for solving LM equations isto transform the matrix equations into systems of linear equations using the Kronecker product Jodar and Mariton (1987) . In this way, various methods for solving linear systems can be applied to the resulting matrix equation. However, because this approach involves Kronecker product operations, its computational cost increases significantly for large-scale matrices. Therefore, iterative methods provide an effective alternative for solving LM equations. Several gradient-based iterative algorithms (Ding and Chen (2005); Zhou et al.(2010)) have been developed for solving LM equations. An implicit iterative algorithm for solving SLM equation was proposed in Zhang et al. (2017) . An iterative algorithm for ⋆ This work was supported by the Guangdong Basic and Applied Basic Research Foundation of China (No. 2026A1515012144) . ∗Corresponding author [chenxs@gdut.edu.cn](chenxs@gdut.edu.cn) (X. Chen) discrete periodic LM equation was proposed in Wu et al.(2018) . For Markovian jump LM equations, Tian et al.(2018) proposed an inner–outer (IO) iterative algorithm for solving the CCMJLM equation. By applying multi-step Smith iterations to the inner iteration, Tian et al. (2020) developed a multi-step Smith IO algorithm for solving the CCMJLM equation. He and Chen (2025) proposed a Jacobi gradient-based iterative algorithm for solving the complex conjugate and transpose Sylvester matrix equation. T

FAQs

What problem does the proposed inner–outer (IO) iteration algorithm address?

It addresses solving a stochastic Lyapunov matrix equation associated with a discrete-time stochastic linear system, aiming to compute the unknown matrix in the Lyapunov formulation.

How does the proposed method improve convergence performance compared with earlier approaches?

It introduces an additional adjustable parameter—the number of inner iteration steps—and develops optimal parameter-selection strategies, which numerical examples show leads to better convergence than several existing iterative methods.

Jul 16, 2026

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