Linear Quadratic differential games and their Open-Loop Nash Equilibrium (OL-NE) strategies are studied with a threefold objective. First, it is shown that the state/costate lifted system (arising from the application of Pontryagin’s Minimum Principle) is such that its behaviour restricted to the equilibrium subspace can be interpreted as the (non-power-preserving) interconnection of two cyclo-passive Port-Controlled Hamiltonian systems. Such PCH systems constitute the best response generators for each player, thus mimicking and extending the corresponding interpretation of (single-player) optimal control problems. Second, by realizing that the behaviour of the lifted dynamics off the equilibrium subspace is “irrelevant” for generating the equilibrium strategies, it is shown that such an invariant subspace can be rendered, via a suitably constructed virtual input, externally asymptotically stable while preserving the OL-NE. Finally, based on these premises we provide a closed-form gradient-descent method to solve the asymmetric coupled Riccati equations characterizing the OL-NE strategies.

中文翻译：

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LQ 差分博弈的 OL-NE：端口控制哈密顿系统视角和一些计算策略


线性二次微分博弈及其开环纳什均衡 （OL-NE） 策略以三重目标进行研究。首先，它表明状态/共态提升系统（由庞特里亚金最小原理的应用产生）是这样的，以至于它的行为仅限于平衡子空间，可以解释为两个循环被动端口控制哈密顿系统的（非守电）互连。这种 PCH 系统构成了每个玩家的最佳响应生成器，从而模拟并扩展了对（单人）最优控制问题的相应解释。其次，通过意识到从平衡子空间中提取的动力学的行为与生成平衡策略“无关”，表明这种不变的子空间可以通过适当构建的虚拟输入呈现，在保持 OL-NE 的同时在外部渐近稳定。最后，基于这些前提，我们提供了一种封闭式梯度下降方法来求解表征 OL-NE 策略的非对称耦合 Riccati 方程。