Covariance Steering for Discrete-Time Linear-Quadratic Stochastic Dynamic Games

This paper addresses the problem of steering a discrete-time linear dynamical system from an initial Gaussian distribution to a final distribution in a game-theoretic setting. One of the two players strives to minimize a quadratic payoff, while at the same time tries to meet a given mean and covariance constraints at the final time-step. The other player maximizes the same payoff, but it is assumed to be indifferent to the terminal constraint. At first, the unconstrained version of the game is examined, and the necessary conditions for the existence of a saddle point are obtained. We show that obtaining a solution for the one-sided constrained dynamic game is not guaranteed, and subsequently the players' best responses are analyzed. Finally, we propose to numerically solve the problem of steering the distribution under adversarial scenarios using the Jacobi iteration method.