Covariance steering in zero-sum linear-quadratic two-player differential games

Abstract: We formulate a new class of two-person zero-sum differential games, in a stochastic setting, where a specification on a target terminal state distribution is imposed on the players. We address such added specification by introducing incentives to the game that guides the players to steer the join distribution accordingly. In the present paper, we only address linear quadratic games with Gaussian target distribution. The solution is characterized by a coupled Riccati equations system, resembling that in the standard linear quadratic differential games. Indeed, once the incentive function is calculated, our problem reduces to a standard one. Tthe framework developed in this paper extends previous results in covariance control, a fast growing research area. On the numerical side, problems herein are reformulated as convex-concave minimax problems for which efficient and reliable algorithms are available.