Paper 'Near-Optimal Mixed Strategy for Zero-Sum Linear-Quadratic Differential Games' accepted to IEEE Transactions on Automatic Control

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. We construct a surrogate pure-strategy stochastic differential game (SDG) by matching the first two moments of the mixed strategies, achieving an $O(\bar{\pi}^{2})$ weak approximation of state distributions and expected costs with respect to the maximum commitment delay $\bar{\pi}$. By analytically resolving the surrogate SDG, we derive closed-form optimal control laws for the matched moments, revealing that the surrogate game is governed by a Generalized Riccati Differential Equation (GRDE) that 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, and certify that both the global value approximation error and the strategy suboptimality gaps are bounded by $O(\bar{\pi}^{1/2})$. Numerical experiments on a double-integrator pursuit–evasion game illustrate the induced physical behaviors and validate the theoretical bounds.