Near-Optimal Mixed Strategy for Zero-Sum Linear-Quadratic Differential Games
This work provides a near-optimal solution for a class of zero-sum differential games, which is an incremental step towards solving complex control problems for researchers in optimal control and game theory.
This paper tackles the challenge of finding optimal mixed strategies in zero-sum linear-quadratic differential games (ZSLQDGs), which previously lacked analytic solutions. The authors synthesize near-optimal mixed strategies by constructing a surrogate stochastic differential game, achieving an $\\mathcal{O}(\ar\\pi^2)$ weak approximation of state distributions and expected costs, and bounding the global value approximation error and strategy suboptimality gaps by $\\mathcal{O}(\ar\\pi^{\ rac{1}{2}})$.
Deriving analytic solutions for optimal mixed strategies in zero-sum linear-quadratic differential games (ZSLQDGs) remains an open problem. In this paper, we analytically synthesize near-optimal mixed strategies for ZSLQDGs and establish rigorous performance certifications. Specifically, we construct a surrogate pure-strategy stochastic differential game (SDG) by matching the first two moments of the mixed strategies. This method achieves an $\mathcal{O}(\barπ^2)$ weak approximation of state distributions and expected costs with respect to the maximum commitment delay $\barπ$. By analytically resolving the surrogate SDG, we derive closed-form optimal control laws for the matched moments. Crucially, we reveal that the surrogate game is governed by a Generalized Riccati Differential Equation (GRDE), which explicitly dictates a dynamic energy allocation law for variance injection. Building on these solutions, we propose a robust dual-routing architecture to execute the near-optimal mixed strategies. Furthermore, we certify that both the global value approximation error and the strategy suboptimality gaps are bounded by $\mathcal{O}(\barπ^{\frac{1}{2}})$. Finally, numerical experiments on a double-integrator pursuit-evasion game illustrate the induced physical behaviors and validate the theoretical bounds.