Differential game with a change of dynamics and control integral constraint of the first player

The one-type game is considered in which the first player has a change of the dynamics and an integral constraint on the control, and the second one has a geometrical constraint. The purpose of the first player is the $\varepsilon$-capture of the enemy at a fixed time $p$ with a minimum expenditure of reserve resources, the goal of the second one is opposite. A necessary and sufficient condition for the solving of the problem by the first player is found.