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FlipDyn，一种两个对立方（防御者与攻击者）通过策略选择最优接管涉及动力系统的资源的动态博弈模型。在每个时间点，玩家可接管资源并控制该动力系统，同时产生状态相关和控制相关的成本。由此形成的模型成为混合动力系统，其离散状态（FlipDyn状态）决定了哪个玩家掌握资源控制权。我们的目标是计算此动态零和博弈的纳什均衡。本文贡献分为四点：第一，针对任意非负成本，给出FlipDyn博弈鞍点值的解析表达式及相应的纳什均衡（NE）接管策略；第二，针对连续状态、二次成本的线性动力系统，建立该博弈在线性状态反馈策略空间存在NE的充分条件；第三，针对标量系统的二次成本，导出了不依赖于动力系统连续状态的NE接管策略与鞍点值；第四，针对高维线性动力系统的二次成本，推导了近似NE接管策略与控制策略，可计算各接管状态下博弈值函数的界。我们通过数值算例（包含对抗者存在时线性动力系统的控制问题）验证了所得结论。