Sufficient solvability conditions for the problem of pursuit under an impulse action

The article considers a linear differential pursuit game under the condition that an integral constraint is imposed on the evader’s control and the pursuer uses an impulse control. These impulse actions on the object are made at predetermined time points, and the corresponding control is represented using the Dirac delta function. The article studies linear conflicts described by a system of ordinary differential equations whose trajectories have jumps at certain times. The terminal set is represented by a cylinder in an $n$-dimensional Euclidean space. The problem is solved using the resolving function method. The fact that the lower bound is reached is proved using the theory of support functions. As a result, the quasi-strategy is replaced by an almost stroboscopic strategy and a method for building this strategy is presented. An example of a nonlinear right-hand side is given.