This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.
翻译:本文件介绍了数字方法(广泛使用的Euler-Maruyama方法)的网络物理系统(CPS),并确立了CPS的理论基础,该理论改变了我们理解数字方法与基本动态系统之间关系的方法。在文献中,它们被视为由不平等联系在一起的单独系统,而CPS则是SDE和数字方法的无缝结合,我们构建了一种新的和一般的Stuchaticive阻塞性差异方程式(SiDEs),可以作为CPS的数值方法的卡通形式。通过CPS方法,我们展示了SDE稳定性与使用Lyapunov稳定性理论确定数字方法的稳定性之间的等同和内在关系。我们为我们的SiDE类开发了Lyapunov稳定性理论,我们运用了我们既定的理论,我们介绍了CPS Lyapunov的不平等,这是Euler-Maruyama方法的CPS(S-Maruyama方法)系统稳定手段的必要和充分条件。我们提出了对线性SDE系统及未来热度工作问题的CPS理论和研究。