具有不稳定子系统的切换随机时滞系统的稳定性研究

In practice, it might be difficult to ensure all subsystems of switched system are stable. Thus, it is necessary to investigate the stability of switched systems with some unstable subsystems. The existing literature usually assumes that subsystems of switched system are invariably stable or invariably unstable. However, if the Lyapunov function is increasing in countable intervals, which implies that the corresponding subsystem has some unstable intervals. Then the stability of the subsystem is time-varying. In fact, if the time for the switched system run on the subsystem contains in a stable interval, the subsystem is stable, otherwise the subsystem is probably unstable. Thus, the stability of the subsystem will change as the time for the switched systems running on the subsystem changes. . In this project, not only the stability of switched systems with some invariably unstable subsystems but also the stability of switched systems with time-varying unstable subsystems will be concerned under the generalized Halanay conditions. This project has some theoretical value, which can provide a theoretical reference for similar research. In addition, this project can offer some theoretical guidance for the practical applications of switched system.

在现实生活中很难保证切换系统的每一个子系统都稳定，因而研究具有部分不稳定子系统的切换系统稳定性是非常必要的。现有文献中通常假定切换系统的子系统恒稳定或恒不稳定。但若子系统的Lyapunov函数在可列个小区间上递增时，子系统具有可列个不稳定的小区间。此时，子系统的稳定性具有时变性，它随着系统切换到子系统的时间点以及在子系统上运行时间长短的改变而改变。本项目将在广义Halanay条件下，研究具有恒不稳定子系统的切换随机时滞系统的稳定性。同时，还将考虑具有时变不稳定子系统的切换随机时滞系统的稳定性。本项目的研究成果可以为同类研究提供新的理论分析方法，具有较为重要的学术价值，并可为切换系统的实际应用提供一定的理论指导。

本项目研究了切换随机时滞系统的稳定性，分别考虑了当切换系统的子系统都稳定与切换系统具有部分不稳定子系统时，切换随机时滞系统的稳定性。同时，本项目还考虑了在时滞脉冲作用下，随机时滞微分系统的稳定性。分别给出了当随机时滞系统稳定时，受到时滞脉冲的扰动情况下，随机时滞系统稳定的充分条件。此外，也研究了当随机时滞系统不稳定时，在时滞脉冲的作用下，随机时滞系统达到稳定的充分条件。并给出数值解的例子验证了以上结论的有效性。