Stability of cyber-physical systems of numerical methods for stochastic differential equations: integrating the cyber and the physical of stochastic systems

Abstract

This paper proposes a novel framework for theory of cyberphysical systems (CPS) as well as numerical study of stochastic differential equations (SDEs), which formulates a hybrid system with stochastic impulsive differential equations (SiDEs) to descibe the integrated dynamics of the physical and the cyber parts of the system that are represented by the exact and numerical solutions of an SDE, respectively, unlike in the literature where they are linked by inequalities. This systematic representation provides not only a holistic view but also a seamless, fully synergistic integration of the cyber subsystem/the numerical method with the physical subsystem/the SDE. To establish a foundation of the proposed framework, this paper initiates the study of a new and general class of SiDEs and develop by Lyapunov methods a stability theory for the general class of SiDEs. Applying the established theory, this paper studies the test problem (Q1) how to reproduce the pth moment exponential stability of the underlying physical system/SDE in the cyber counterpart/the widely-used Eular-Maruyama method and the converse (Q2) whether the pth moment exponential stability of the underlying physical system/SDE can be inferred from that of the cyber counterpart/the numerical method. The proposed theory removes the principal barrier to developing CPS and establishes a new systems science that comprehends cyber and physical resources in a single unified framework. As an application, this paper presents the cyber-physical/numerical Lyapunov-Itô inequality for linear systems, which is the necessary and sufficient condition for mean-square exponential stability of the physical system/the SDE, its cyber counterpart/the Eular-Maruyama method, and the resulting CPS/SiDE. This work initiates the studies of systems numerics and the systems science for CPS, in which there are many open and challenging problems.