【作者】 王晓辉；

【导师】 王玲娣；

【作者基本信息】 河南大学 ， 统计学， 2020， 硕士

【摘要】 最近,Cao等学者(2019)考虑利用线性离散时间噪音反馈Markov切换下微分方程并且通过构造李雅普诺夫函数的方法给出其渐进稳定的充分条件.他们研究的方程没有考虑到泊松跳的影响,本文将考虑利用离散观测值产生的噪音反馈随机微分方程并给出其p阶矩指数稳定的判别条件.针对状态依赖切换下随机延迟微分方程,本文首先利用Pei&Xu(2017)中的结论给出该方程系数在非利普希茨条件下解的存在性和唯一性的条件;然后通过Xi&Yin(2013)中状态依赖的切换过程与齐次Markov过程的联系,利用M矩阵等理论知识得到了该方程p阶指数稳定的充分条件;最后推广Shao&Xi(2019)的结果给出其p阶矩指数稳定的判别条件并举例对结论进行验证.针对齐次Markov切换下随机延迟微分方程,本文首先利用Zhao&Zhang(2017)的思路构造满足p阶矩指数稳定的辅助方程来给出该方程p阶矩指数稳定的判别条件;然后通过改进Liu(2016)的向量李雅普诺夫函数给出该方程均方有界的充分条件;最后结合向量李雅普诺夫函数与Wu&Liu(2017)中多重李雅普诺夫函数之间的联系给出该方程均方指数稳定的充分条件并举例对结果进行验证.更多还原

【Abstract】 Recently,Cao et al(2019)had constructed the Lyapunov function to study the asymptotic stability of stochastic switching differential equations driven by linear discrete time noises,but which they studied did not take into account poisson jump processes.In this paper,we study the pth moment exponential stability of stochastic differential equations by feedback control based on the noises of the discrete-time observations.For stochastic differential equations with state-dependent switching,the existence and u-niqueness of its solution are studied by using the conclusions of Pei&Xu in 2017,whose coefficients satisfy the non-lipschitz conditions,then,the sufficient conditions for its pth mo-ment exponential stability are received by combining M Matrix with the relations of the state-dependent switching process and the homogeneous Markov process given by Xi&Gin in 2013,finally,the criteria of its pth moment exponential stability are obtained by the results of Shao&Xi in 2019 and some examples are presented to verify these results.For stochastic differential equations with Markov switching,its pth moment exponential stability is studied by constructing auxiliary equations with help of Zhao&Zhang’s ideas in 2017,which satisfy the pth moment exponential stability,then,its asymptotic Boundedness in mean square is studied by improving the vector Lyapunov function given by Liu in 2016,finally,the sufficient conditions of its exponential stability in mean square are obtained by the Vector Lyapunov functions and the Multiple Lyapunov function given by Wu&Liu in 2017 and some examples are presented to verify these results.更多还原