<p style='text-indent:20px;'>We study the Hamilton-Jacobi (HJ) approach for a two-person zero-sum differential game with state constraints and where controls of the two players are coupled within the dynamics, the state constraints and the cost functions. It is known for such problems that the value function may be discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. In this work, we characterize this value function through an auxiliary differential game free of state constraints. Furthermore, we establish a link between the optimal strategies of the constrained problem and those of the auxiliary problem and we present a general approach allowing to construct approximated optimal feedbacks to the constrained differential game for both players. Finally, an aircraft landing problem in the presence of wind disturbances is given as an illustrative numerical example.</p>

中文翻译：

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具有状态约束的微分博弈控制问题

<p style='text-indent:20px;'>我们研究了具有状态约束的两人零和微分博弈的 Hamilton-Jacobi (HJ) 方法，其中两个玩家的控制在动态中耦合，状态约束和成本函数。对于此类问题，已知值函数可能是不连续的，并且通过 HJ 方程对其进行表征需要一些涉及动力学和状态约束集的可控性假设。在这项工作中，我们通过无状态约束的辅助微分博弈来表征这个价值函数。此外，我们在约束问题的最优策略和辅助问题的最优策略之间建立了联系，并且我们提出了一种通用方法，允许为双方玩家构建约束微分博弈的近似最优反馈。最后，给出了一个存在风扰的飞机着陆问题作为说明性数值例子。</p>