Linear Quadratic Nash Differential Games of Stochastic Singular Systems

1 Introduction

Singular systems, also known as descriptor systems, generalized state-space systems and implicit systems are described by differential-algebraic equations. Singular systems have been extensively studied over the past decades due to the fact that they can describe a great many natural phenomena in physical systems such as microelectronic circuits, economics, demography and so on[1–4].

As is well known, environmental noise exists and cannot be neglected in many dynamical systems[5–9]. And there are some results about the research of LQ control for stochastic singular systems: Balasubramaniam and Kumaresan[10] discussed a solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular system using neural networks. The optimal control problem for a class of stochastic LQ singular periodic neuro Takagi-Sugeno fuzzy systems by using ant colony programming was studied in [11]. Kumaresan and Kuru[12] investigated the optimal control problem for stochastic linear singular Takagi-Sugeno fuzzy delay systems with quadratic performance using genetic programming. Zhang and Xing[13] investigated the stability and optimal control for stochastic singular systems.

On the other hand, during the past decades, differential games have been playing a central role in economy, ecology and elsewhere, such as optimizing behavior and enduring consequences of decisions. For that reason, this framework has developed into a major research field in control theory, and has many applications for solving real world problems, see [14–16]. The differential games of singular systems can describe many practical situations and have been widely studied by many researchers for several decades. Among these, most researchers studied the differential games of deterministic singular systems in finite-time horizon and infinite-time horizon, we refer the reader to [17–21] and the references therein.

However, to the best of our knowledge, few results have been obtained of the differential games for stochastic singular systems. Motivated by this, we deal with the LQ Nash differential games of stochastic singular systems.

It is worth mentioning that many researchers discussed the Nash differential games of singular systems by disintegrating the systems into two: fast and slow[18, 19, 21]. Recently Zhang and Xing deal with the stochastic singular systems as a whole by using the square completion technique, and get the results for optimal control problems involving new stochastic generalized Riccati equation. Inspired by this, we apply the same method to the corresponding differential game problems and get a set of cross-coupled stochastic Riccati equations. Actually, the conclusion is more intuitive and calculation is more easily.

The rest of the paper is organized as follows: Some important definitions and lemmas which are useful to obtain the main results are introduced in section 2. The differential games of stochastic singular system in finite-time horizon are discussed in section 3. In this case, the problem is formulated firstly. Then, the corresponding cross-coupled Riccati differential equations are derived and the existence of their solutions is shown to be sufficient conditions for the Nash strategies. The differential games of stochastic singular system in infinite-time horizon discussed in section 4. Similarly, the corresponding cross-coupled Riccati algebraic equations are derived and the existence of their solutions is shown to be sufficient conditions for the Nash strategies. In section 5, the conclusions are given.

For convenience, we will make use of the following notations throughout this paper:

A′ : transpose of a matrix or vector A; A⁻¹: inverse of a matrix or vector A; A > 0 (A ≥ 0): positive definite (positive semi-definite) matrix A; ε[⋅]: the expectation operator with respect to the given probability measure P. Rⁿ: the n-dimensional Euclidean space; S^{n × n} : the set of all n × n matrices; Sⁿ: the space of all n × n symmetric matrices; S+n : the space of all nonnegative definite matrices of Sⁿ; rank(A): the rank of A; deg(det(sI − A)): degree of determinant sI − A.

2 Preliminaries

In this section, we introduce some important definitions and lemmas which are useful to obtain our main results later.

Throughout this paper, let (Ω, F, {F_t}_{t ≥ 0}, P) be a given filtered probability space where there exists a standard one dimensional Wiener process {w(t)}_{t ≥ 0}. Consider the following stochastic singular systems

where x₀ ∈ Rⁿ is the initial state of the systems, A ∈ R^{n × n}, C ∈ R^{n × n} are known coefficient matrices associated with x(t), respectively. E is a known singular matrix with rank(E) = r ≤ n.

3 Stochastic Nash differential games in finite-time horizon

4 Stochastic Nash differential games in infinite-time horizon

5 Conclusions

In this paper, we dealt with the stochastic Nash games in finite-time horizon and infinite-time horizon, respectively. Basing on the results of the stochastic optimal control problem, the Nash strategies for stochastic singular systems are derived by the method of a set of cross-coupled stochastic Riccati equations. We note that the strategies of the two players are interacted and it has applications for solving real world problems. For example, considering the investment stock problem of a security company, both the investors and security company need to consider the strategies of each other when they choose the strategies. Hence, it would be necessary to consider the differential games for the stochastic singular systems to obtain an equibrilium. Also the results are easily obtained from those of LQ scochastic singular systems, the method and the stochastic cross-coupled Raccti equations are new for the Nash games of the scochastic singular system. Due to various reasons, we only deal with the open-loop Nash equilibrium for the stochastic singular systems, and the close-loop Nash equilibrium is the next step of research.