  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.
