脉冲随机微分系统的噪声镇定和噪声消稳研究

Since noise can be used to destabilize a given stable differential system and it can also be used to stabilize a given unstable system or to make a given system whose solutions grow exponentially become a new system whose solutions will grow at most polynomially,this project mainly focuses on the stabilization and destabilization of impulsive stochastic differential systems by noise. To be specific, for linear systems, based on the Lyapunov stability theory, we will estabilishe some criterion on the almost sure exponential stability/ instability by using the ergodic theorem and the semigroup theorem. And for the general nonlinear systems, by utilizing the It? formula, Doob martingale exponential inequalities and the Borel-Cantelli Lemma, the general theorem on the almost sure exponential stability/instability will be proposed, and some sufficient conditions on the stability will also be given in terms of LMIs. Furthermore, based on the stability results,the exact method how to design a noise control to stabilize/destabilize a given linear or nonlinear impulsive differential system will be discussed.. The researches of this project will rich the stability theory of hybrid stochastic systems, and it may be provide a theoretical approach for people to deeper understanding the influence of stochastic noise on natural systems and engineering systems.

鉴于随机噪声既可以将给定的稳定微分系统扰动成不稳定的，也可以用来镇定不稳定的微分系统或使给定微分系统的解由指数增长变成最多多项式增长，本课题主要研究线性和非线性脉冲随机微分系统的噪声镇定/噪声消稳问题.具体说来，对线性脉冲随机微分系统，基于Lyapunov稳定性理论，利用遍历理论和算子半群理论建立其几乎必然指数稳定/不稳定的判据；对一般的非线性脉冲随机微分系统，利用It?公式、Doob鞅指数不等式和Borel-Cantelli引理等随机分析技巧建立其几乎必然指数稳定/不稳定的一般性定理，并给出系统基于LMIs形式的稳定性充分条件；在稳定性研究的基础上进而给出线性和非线性脉冲微分系统噪声镇定/噪声消稳的具体设计方法.. 本课题的研究将丰富混杂随机系统稳定性理论，为人们更深入地认识随机噪声对自然系统和工程系统的影响规律提供理论方法.

鉴于随机噪声和脉冲扰动既可以将给定的稳定微分系统扰动成不稳定的，也可以用来镇定不稳定的微分系统，本课题主要研究了一般形式的线性和非线性脉冲随机系统的几乎必然指数稳定性以及噪声镇定问题。首先，我们基于Lyapunov 稳定性理论，利用Itô 公式、Doob 鞅指数不等式、B-D-G不等式和Borel-Cantelli 引理等随机分析技巧建立其几乎必然指数稳定/不稳定的一般性定理；在稳定性研究的基础上进而给出系统系统噪声镇定/噪声消稳的具体设计方法.；最后，将所得结果应用于线性系统，得到了系统均方指数稳定及a.s.指数稳定的LMIs形式的判据。