We present the FlipDyn, a dynamic game in which two opponents (a defender and an adversary) choose strategies to optimally takeover a resource that involves a dynamical system. At any time instant, each player can take over the resource and thereby control the dynamical system after incurring a state-dependent and a control-dependent costs. The resulting model becomes a hybrid dynamical system where the discrete state (FlipDyn state) determines which player is in control of the resource. Our objective is to compute the Nash equilibria of this dynamic zero-sum game. Our contributions are four-fold. First, for any non-negative costs, we present analytical expressions for the saddle-point value of the FlipDyn game, along with the corresponding Nash equilibrium (NE) takeover strategies. Second, for continuous state, linear dynamical systems with quadratic costs, we establish sufficient conditions under which the game admits a NE in the space of linear state-feedback policies. Third, for scalar dynamical systems with quadratic costs, we derive the NE takeover strategies and saddle-point values independent of the continuous state of the dynamical system. Fourth and finally, for higher dimensional linear dynamical systems with quadratic costs, we derive approximate NE takeover strategies and control policies which enable the computation of bounds on the value functions of the game in each takeover state. We illustrate our findings through a numerical study involving the control of a linear dynamical system in the presence of an adversary.

翻译：本文提出了FlipDyn，一种动态博弈模型，其中两个对手（防御者和攻击者）选择策略以最优方式接管一个涉及动态系统的资源。在任意时刻，每个玩家均可接管资源，从而控制该动态系统，但需承担状态依赖和控制器依赖的成本。该模型成为一个混合动力系统，其离散状态（FlipDyn状态）决定了当前由哪一方控制资源。我们的目标是计算这一动态零和博弈的纳什均衡。本文贡献包括四个方面：首先，针对任意非负成本，给出了FlipDyn博弈鞍点值的解析表达式及相应的纳什均衡接管策略；其次，对于具有二次成本的连续状态线性动态系统，建立了在线性状态反馈策略空间中博弈存在纳什均衡的充分条件；第三，针对具有二次成本的标量动态系统，推导出与动态系统连续状态无关的纳什均衡接管策略和鞍点值；最后，对于具有二次成本的高维线性动态系统，推导了近似纳什均衡接管策略与控制策略，从而能够计算各接管状态下博弈值函数的边界。我们通过一个包含攻击者控制线性动态系统的数值案例验证了上述结论。