Linear differential games with impulse control of players

We consider pursuit games in which players (the pursuer, evader or both players) use impulse controls, which is expressed in terms of the Dirac delta-function. We study linear dynamical system described by ordinary differential equations whose trajectories have discontinuities at discrete time instants. Such systems are a kind of hybrid ones. The research is based on the principal ideas of the decision function method. The following cases are considered successively: impulse control of the pursuer; impulse control of the evader; impulse control of both players. For each of the three cases, the problem of approach to a cylindrical terminal set is studied and sufficient solvability conditions for the above problem are obtained. The theoretical results are illustrated with an example of a pursuit game with simple motion.