针对随机控制系统数值解的均方指数输入状态稳定性

分析了随机控制系统数值解的均方指数输入状态稳定性. 首先, 针对随机控制系统, 随机θ-方法满足有限时间强收敛条件. 然后, 我们证实, 在有限时间强收敛条件下, 随机控制系统是均方指数输入状态稳定的当且仅当随机θ-方法(充分小步长)是均方指数输入状态稳定的. 另外, 对一类满足单边Lipschitz条件的随机控制系统, 有两类隐式欧拉方法(对任意步长)能够继承原系统的均方指数输入状态稳定性. 最后, 一些数值实例证实了本文所获结论的正确性.

Mean-Square Exponential Input-to-State Stability of Numerical Solutions for Stochastic Control Systems

This paper deals with the mean-square exponential input-to-state stability (exp-ISS) of numerical solutions for stochastic control systems (SCSs). Firstly, it is shown that a finite-time strong convergence condition holds for the stochastic θ-method on SCSs. Then, we can see that the mean-square exp-ISS of an SCS holds if and only if that of the stochastic θ-method (for suffciently small step sizes) is preserved under the finite-time strong convergence condition. Secondly, for a class of SCSs with a one-sided Lipschitz drift, it is proved that two implicit Euler methods (for any step sizes) can inherit the mean-square exp-ISS property of the SCSs. Finally, numerical examples confirm the correctness of the theorems presented in this study.