  In this paper, we formulate a two-player zero-sum game under dynamic
constraints defined by hybrid dynamical equations. The game consists of a
min-max problem involving a cost functional that depends on the actions and
resulting solutions to the hybrid system, defined as functions of hybrid time
and, hence, can flow or jump. A terminal set conviniently defined allows to
recast both finite and infinite horizon problems. We present sufficient
conditions given in terms of Hamilton-Jacobi-Bellman-Isaacs-like equations to
guarantee to attain a solution to the game. It is shown that when the players
select the optimal strategy, the value function can be evaluated without
computing solutions to the hybrid system. Under additional conditions, we show
that the optimal state-feedback laws render a set of interest asymptotically
stable for the resulting hybrid closed-loop system. Applications of this
problem, as presented here, include a disturbance rejection scenario for which
the effect of the perturbation is upper bounded, and a security scenario in
which we formulate an optimal control problem under the presence of a
maximizing attack.
