Differential games with impulse control

This chapter deals with the pursuit games in which players (pursuer, evader, or both) employ impulse control. We consider continuous-time dynamical systems modeled by ordinary differential equations that are affected by jumps in state at discrete instants. The moments of jump comply with the condition for a finite number of jumps in finite time. In so doing, the Dirac delta function is used to describe the impulse control. Such systems represent a special case of hybrid systems. The method of resolving functions provides a general framework for analysis of the above-mentioned problems. This method essentially uses the technique of the theory of set-valued mappings. The following cases are examined in succession: impulse control of the pursuer; impulse control of the evader; impulse control of both players. The problem of approaching a cylindrical terminal set is studied for each case, and the sufficient conditions for its solvability are derived. Obtained results are supported by a model example of the game with simple motion dynamics.