Closed-Loop Nash Equilibrium in the Class of Piecewise Constant Strategies in a Linear State Feedback Form for Stochastic LQ Games

Abstract

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach.

Keywords

• nash equilibria
• stochastic LQ differential games
• sampled-data controls
• equilibrium strategies
• optimal trajectories