Impulse differential game with a mixed constraint on the choice of the control of the first player

We consider a linear differential game in which the first player can choose both an impulse control and a control subject to a geometric constraint. The first player can use a prescribed amount of resource to form the impulse control. Portions of this amount can be separated at certain times, thus producing “instantaneous” changes of the state vector and complicating the problem. The control of the second player is subject to a geometric constraint. The vectograms of the players are described by the same ball with different time-dependent radii. The terminal set is a ball with fixed radius. The aim of the first player is to bring the state vector to the terminal set at a given time. The aim of the second player is opposite. Necessary and sufficient conditions for meeting the terminal set at the given time are found, and the corresponding controls of the players guaranteeing the achievement of their goals are constructed. A solution of an example illustrating the theory is given.