A Game—Theoretic Model for a Stochastic Linear Quadratic Tracking Problem

• Published in: Axioms Vol. 12; no. 1; p. 76
• Main Authors: Drăgan, Vasile, Ivanov, Ivan Ganchev, Popa, Ioan-Lucian
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.01.2023
• Subjects: Algebra; Cooperation; Decision making; Differential games; Dynamical systems; Equilibrium; Game theory; Hybrid systems; Linear equations; linear quadratic tracking problem; Markov analysis; Mathematical models; Nash equilibria; Nonlinear differential equations; Nonlinear equations; Perturbation; Process controls; Random variables; Reference signals; stochastic linear differential games; Stochastic models; Strategy; Tracking control; Tracking problem
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms12010076

In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations.