Stochastic linear quadratic differential games in a state feedback setting with sampled measurements

Abstract

The problem of sampled-data Nash equilibrium strategy in a state feedback setting for a stochastic linear quadratic differential game is addressed. It is assumed that the admissible strategies are constant on the interval between two measurements. The original problem is converted into an equivalent one for a linear stochastic system with finite jumps. This new formulation of the problem allows us to derive necessary and sufficient conditions for the existence of a sampled-data Nash equilibrium strategy in a state feedback form. These conditions are expressed in terms of solvability of a system of interconnected matrix linear differential equations with finite jumps and subject to some algebraic constraints. We provide explicit formulae of the gain matrices of the Nash equilibrium strategy in the class of piece-wise constant strategies in a state feedback form. The gain matrices of the feedback Nash equilibrium strategy are computed based on the solution of the considered system of matrix linear differential equations with finite jumps. For the implementation of these strategies only measurements at discrete-time instances of the states of the dynamical system are required. Finally, we show that under some additional assumptions regarding the sign of the weights matrices in the performance criteria of the two players, there exists a unique piecewise Nash equilibrium strategy in a state feedback form if the maximal length of the sampling period is sufficiently small.