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

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

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.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.
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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