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Article · Journal of Industrial and Management Optimization · January 1, 2025

An iterative method for solving $ \mathrm{It\hat{o}} $ Lyapunov matrix equations associated with $ \mathrm{It\hat{o}} $ -type stochastic systems

Hui‐Jie Sun, Fangzhou Fu, Muhammad Imran, Yuyao Wu

Muhammad Imran
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Abstract

This paper addresses the problem of solving a type of Lyapunov matrix equation that is associated with stochastic systems in the continuous-time domain. A novel iterative method is developed to find the unique solution of the considered Lyapunov matrix equation in the explicit form. In comparison with some of the existing iterative methods, the method developed in this paper has a simpler structure and faster convergence speed. By analysing the spectral radius of the corresponding iteration matrix, we can present a set of sufficient and necessary conditions for the convergence of the presented explicit iterative method. A simple method is then given for selecting the optimal tuning parameter, which can ensure that the proposed iterative method gets the fastest convergence speed. The validity of the explicit iteration algorithm is verified by means of two examples.

Matrix Theory and AlgorithmsIterative Methods for Nonlinear EquationsNumerical methods for differential equationsstochastic systemscontinuous-time stochastic systemsiterative methods

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