Nash strategy of multiparameter singularly perturbed Markov jump stochastic systems with state- and control-dependent noise

Abstract

This paper investigates Nash games for a class of multiparameter singularly perturbed stochastic systems governed by Itô's differential equation with Markov jump parameters with state-and control-dependent noise. First, in order to obtain Nash equilibrium strategies, cross-coupled stochastic algebraic Riccati equations (CSAREs) are introduced. After establishing an asymptotic structure with positive definiteness for solutions of CSAREs, feasible numerical algorithms by means of the linear matrix inequality (LMI) are considered. In addition, the mode-independent Nash strategy set is also investigated. Finally, we provide a numerical example to verify the efficiency of the proposed algorithms.

Optimal control for a singularly perturbed linear stochastic system with multiplicative white noise perturbations and Markovian jumping

“…In order to overcome this inconvenience, in this work, we make the following assumption: Assumption The small parameters ϵ > 0, μ > 0 satisfy the condition Remark Assumption is fulfilled, for example, if the parameters μ and ϵ satisfy a relation of the form μ = μ ( ϵ ) where μ :[0, ∞ )→[0, ∞ ) is a function with the properties: μ ( ϵ ) = 0 iff ϵ = 0. . . A special case when the aforementioned conditions are fulfilled is (see, e.g., ). On the other hand, in , is assumed. In this case, it should be noted that the asymptotic structure of the solution for , , , , , can be easily derived.…”

“…Because many applications of stochastic systems are governed by an Itô differential equation, the theory of optimal control and the related area for singularly perturbed and multi‐parameter stochastic systems has been well documented . These works have shown that there exists a parameter‐independent reduced‐order control that can achieve an arbitrary approximation of the optimal cost.…”

“…Recently, necessary and sufficient conditions for the stabilizability of multi‐parameter singularly perturbed stochastic systems (MSPSS), which can be viewed as an extension of , were obtained in the form of linear matrix inequalities (LMIs) . On the other hand, the Nash games for a class of Markovian jump linear systems have been investigated . It is well known that these systems are a class of stochastic hybrid systems.…”

“…Although the MSPSS has been studied in , the existence of the stabilizing solution has not been discussed in detail.…”

“…In general, because the two‐time scale approach is the alleviation of the high dimensionality and ill‐conditioning resulting from the interaction of slow and fast dynamic modes, another approach is necessary. It should be noted that while the Nash games for the MSPSS with Markovian jumping parameters have been studied in , a strong assumption for the coefficient matrix was imposed to attain an easy derivation. That is, because the slow subsystems have no state and control‐dependent noise, the difficulties of analysis and design could be avoided easily.…”