Pursuit-evasion problems under nonlinear increase of the pursuer's resource

In the paper, we investigate pursuit-evasion problems in a simple motion differential game with two players, termed a pursuer and an evader. We put different kinds of non-stationary integral constraints, which restrict the energy consumption rate of the players. On the other hand, it is assumed that at each time the players have some additional amount of control resource. The integral constraint on the control of the pursuer is given under certain conditions, which include a non-stationary integral constraint. Firstly, the reachable set of each player is determined. We put forward the parallel approach strategy, which is known as a $\Pi$-strategy, for the pursuer, and as a result, we get necessary and sufficient conditions of capture. To solve the evasion problem, a specific admissible strategy is provided for the evader and a sufficient condition is obtained. Furthermore, in the pursuit problem, an optimal capture time is found through the strategy of the evader. In order to illustrate the obtained results, several examples are given, where guaranteed capture times are proposed for the pursuit problems and lower bounds for the distances between the players are obtained for the evasion problem. This work extends the results and methods from the works of R.Isaacs, L.A.Petrosjan, N.N.Krasovskii, A.A.Chikrii, A.A.Azamov, and other authors.