具有无限马尔可夫切换的离散时间随机系统的最大值解、最小半正定解与稳定解的研究

The Maximal, Minimal Positive Semidefinite and Stabilizing Solutions for Discrete-Time Stochastic Systems with Infinite Markov Switching

摘要: 本文主要研究具有无限马尔可夫切换的离散时间随机系统的最值解与稳定解。在研究具有有限马尔可夫切换的离散时间随机系统的最值解与稳定解的基础上推广到无限马尔可夫，为研究系统稳定性奠定了良好的理论基础。文章首先介绍了稳定解，最大值解与半正定最小值解的概念，并利用算子理论和随机分析等方法得出系统随机稳定能够等价于相应的正算子序列是稳定的；其次，在系统所对应的Riccati方程解集非空的前提下，若Riccati方程有稳定解，则必定存在最大值解；再次，添加系统随机可探测条件，系统能够存在最小半正定解，若考虑存在唯一稳定解，则系统的最大值解等于系统的稳定解也等于系统的最小半正定解；最后，用数值举例来验证定理的正确性和有效性。

Abstract: In this paper, we study the maximal, minimal positive semidefinite and stabilizing solutions of dis-crete-time stochastic systems with infinite Markov switching. On the basis of studying the optimal solution and stable solution of discrete time stochastic system with finite Markov switching, it is ex-tended to infinite Markov, which lays a good theoretical foundation for the study of system stability. Firstly, the concepts of stabilizing solution, maximal solution and minimal positive semidefinite so-lution are introduced. By using operator theory and stochastic analysis methods, it is shown that the stochastic stability of the system is equivalent to that the corresponding sequence of positive operators is stable. Secondly, on the premise that the solution set of the corresponding Riccati equation is non-empty, if there is a stabilizing solution to the Riccati equation, there must be a maximal solution. Thirdly, under the condition that the system is stochastic detectability, the min-imal positive semidefinite solution can exist. If the unique stable solution is considered, the maxi-mal solution of the system is equal to the stabilizing solution and the minimal positive semidefinite solution. Finally, an example is given to verify its correctness and validity.