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

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.

💡 Research Summary

The paper introduces a novel class of two‑player zero‑sum differential games in a stochastic setting where, in addition to the usual quadratic running costs, a specification on the terminal state distribution is imposed. The authors consider linear‑quadratic (LQ) games with Gaussian target distributions and show how to steer the joint state distribution to the desired terminal covariance by adding an appropriately designed incentive term to the game’s cost.

The starting point is the classical covariance‑control problem: for a linear stochastic system (dx = Ax,dt + Bu,dt + C,dw) with initial covariance (\Sigma_0), one seeks a feedback law that drives the state to a target Gaussian distribution (\mathcal N(0,\Sigma_T)). The optimal feedback is (u = -B^\top \Pi x) where (\Pi(t)) and an auxiliary matrix (H(t)) satisfy a coupled pair of Riccati‑type differential equations (3a‑3d). These equations are closely related to Schrödinger bridge and optimal mass transport formulations.

The paper then extends this setting to a zero‑sum LQ differential game with dynamics
(dx = Ax,dt + B_1 u,dt + B_2 v,dt + C,dw). Player 1 minimizes
(J_1 = \mathbb E!\int_0^T!\big