Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems

Abstract

The Bogoliubov-Mitropolsky small parameter method is used to study the behavior of stochastic differential systems in the analysis of the corresponding properties of solutions of averaged systems. INTRODUCTIONAveraging over explicit time [4] is one of the most popular methods for studying dynamic systems. This method also performs well in the theory of random perturbations [1, 2, 5, 6, 10-13] (see the references therein for a more detailed bibliography).This method allows not only setting up an averaged system that approximately describes the dynamics of the original model, but also writing stochastic differential equations (SDEs) for normalized deviations of the solution of the original equation from the corresponding solutions of averaged motion.The Bogoliubov-Mitropolsky principle of averaging is proved in the monograph [5] for impulsive systems and in [6] for the case of random perturbations. For clarity, let us first present the well-known results from [5,6,14]. In contrast to most studies on the averaging method of impulsive systems, we will consider impulsive systems with Markov switching.In the paper, we will prove that limit theorems can be used for stability analysis as is the case with smooth stochastic systems [10][11][12][13]. In the second and third sections, an averaged system and diffusion approximation are set up; their analysis with the use of modern computer technologies allows impruving the computational efficiency by a factor of tens of millions.

Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations

“…Then, by the definition of the discrete Lyapunov operator (lv k )(y, h, x) (see (7)) and from Equation (11), taking into account (10), we obtain the following inequality:…”

Optimal Control of Stochastic Dynamic Systems of a Random Structure with Poisson Switches and Markov Switching

“…Stability in different probabilistic senses has been investigated, and the problem of optimal stabilization, solution of which is the control stabilizing the system to stochastically stable, is solved. In these papers, the absence of perturbation points was assumed, but in [14][15][16], the existence and uniqueness of the solution of a system of differential-difference equations with Markov parameters and switching in the presence of perturbation points were proved. erefore, we can consider the problem of stability and optimal control for such systems.…”

“…Also, sufficient conditions for the stability of prelimited processes are discussed. In the articles [15,16], Tsarkov et al discussed the influence of Markov perturbation on the solutions and applied stochastic differential equations for applied problems.…”