Asymptotic Stability With Probability One of Random Time Delay Controlled Quasi Integrable Hamiltonian Systems

Abstract

A new procedure for determining the asymptotic stability with probability one of randomtimedelaycontrolled quasiintegrable Hamiltonian systems is proposed. Such a system is formulated as continuous–discrete hybrid system and the random time delay is modeled as a Markov jump process. A threestep approximation is taken to simplify such hybrid system: (i) the randomly periodic approximate solution property of the system is used to convert the random time delay control into the control without time delay but with delay time as parameter; (ii) a limit theorem is used to transform the hybrid system with Markov jump parameter into one without jump parameter; and (iii) the stochastic averaging method for quasiintegrable Hamiltonian systems is applied to reduce the system into a set of averaged Itأ´ stochastic differential equations. An approximate expression for the largest Lyapunov exponent of the system is derived from the linearized averaged Itأ´ equations and the necessary and sufficient condition for the asymptotic stability with probability one of the system is obtained. The application and effectiveness of the proposed procedure are demonstrated by using an example of stochastically driven twodegreesoffreedom networked control system (NCS) with random time delay.