On a control problem with a bound on the number of switchings

A linear differential game of approach and evasion is considered. A special feature of the statement of the problem consists in the fact that the seco nd player (the evader) can form only piecewise constant controls having at most m (m greater than or equal to 1) switchings. This problem was formaliz ed in works of A.G. Chentsov. In particular, it is shown that the decision procedure for the second player is represented by the following triple: the positional strategy, the number of switching, and the correction law that determines the method for processing the trajectory being executed and the choice of the point of the next switching. In this paper, the algorithm for the stepwise execution of motions that correspond to the decision rule of the evader is proposed as a specific example of the differential game under consideration.