Near-Optimal Mixed Strategy for Zero-Sum Linear-Quadratic Differential Games
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911731314327552 |
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| author | Xu, Tao Xi, Wang He, Jianping |
| author_facet | Xu, Tao Xi, Wang He, Jianping |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30886 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Near-Optimal Mixed Strategy for Zero-Sum Linear-Quadratic Differential Games Xu, Tao Xi, Wang He, Jianping Optimization and Control Systems and 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. 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. |
| title | Near-Optimal Mixed Strategy for Zero-Sum Linear-Quadratic Differential Games |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2605.30886 |