FlipDyn: A game of resource takeovers in dynamical systems

We introduce a game in which two players with opposing objectives seek to repeatedly takeover a common resource. The resource is modeled as a discrete time dynamical system over which a player can gain control after spending a state-dependent amount of energy at each time step. We use a FlipIT-inspired deterministic model that decides which player is in control at every time step. A player's policy is the probability with which the player should spend energy to gain control at each time step. Our main results are three-fold. First, we present analytic expressions for the cost-to-go as a function of the hybrid state of the system, i.e., the physical state of the dynamical system and the binary \texttt{FlipDyn} state for any general system with arbitrary costs. These expressions are exact when the physical state is also discrete and has finite cardinality. Second, for a continuous physical state with linear dynamics and quadratic costs, we derive expressions for Nash equilibrium (NE). For scalar physical states, we show that the NE depends only on the parameters of the value function and costs, and is independent of the state. Third, we derive an approximate value function for higher dimensional linear systems with quadratic costs. Finally, we illustrate our results through a numerical study on the problem of controlling a linear system in a given environment in the presence of an adversary.