Single-type problem of impulse pursuit of an object with changing dynamics

We consider a linear differential game with an impulse control of the first player. The capabilities of the first player are determined by the stock of resources that he can use to form his control. 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. It is believed that the second player at a time moment unknown in advance to the first player can change his dynamics once. The terminal set is a ball with a fixed radius. The goal of the first player is to lead the phase vector to the terminal set at a given time. The goal of the second player is the opposite. Necessary and sufficient conditions for a meeting the terminal set at the given time are found. The corresponding controls of the players, guaranteeing the achievement of their goals are constructed.