Stabilization of stochastic dynamical systems of a random structure with Markov switches and Poisson perturbations

TL;DR

This paper develops a novel method for stabilizing and optimally controlling stochastic dynamical systems with random structures, Markov switches, and Poisson perturbations, using Lyapunov functions, Riccati equations, and small parameter techniques.

Contribution

It introduces a new approach for optimal stabilization of complex stochastic systems with random structures, including stability conditions and control synthesis methods.

Findings

Derived sufficient stability conditions using second Lyapunov method.

Proposed a Riccati equation-based method for optimal control synthesis.

Justified a small parameter approach for algorithmic control search.

Abstract

An optimal control for a dynamical system optimizes a certain objective function. Here we consider the construction of an optimal control for a stochastic dynamical system with a random structure, Poisson perturbations and random jumps, which makes the system stable in probability. Sufficient conditions of the stability in probability are obtained, using the second Lyapunov method, in which the construction of the corresponding functions plays an important role. Here we provide a solution to the problem of optimal stabilization in a general case. For a linear system with a quadratic quality function, we give a method of synthesis of optimal control based on the solution of Riccati equations. Finally, in an autonomous case, a system of differential equations was constructed to obtain unknown matrices that are used for the building of an optimal control. The method of a small parameter is justified for the algorithmic search of an optimal control. This approach brings a novel solution to the problem of optimal stabilization for a stochastic dynamical system with a random structure, Markov switches and Poisson perturbations.