The problem of trajectories avoiding a sparse terminal set

The problem of avoidance (evasion) in conflict-controlled processes in L.S. Pontryagin and E.F. Mishchenko’s statement is considered. The terminal set has a special discrete (sparse) structure. In contrast to other works, it consists of a countable number of points with distances not limited from below by a fixed positive constant. New sufficient conditions and an evasion method are obtained which make it possible to solve a number of avoiding trajectory problems for oscillatory systems, including the problem of swinging a generalized mathematical pendulum.