Principle of Time Dilation in Game Problems of Dynamics - Chikrii, G. Ts (Recent Developments in Automatic Control Systems, 2022)
In this chapter, we consider the linear differential game of approaching a cylindrical terminal set. The case when classic Pontryagin's condition does not hold is studied. Instead, a considerably weaker condition, incorporating the function of time dilation, is introduced. This makes it possible to expand the range of problems that can be solved analytically, by using information about the enemy's control in the past. The study is carried out in the framework of the Method of Resolving Functions. The gist of the method consists in constructing certain scalar function associated with the parameters of the conflict-controlled process and characterizing the gain of the first player at each moment. If the total gain achieves the predetermined value then this means that the game terminates in a certain guaranteed time. The pursuer's control, realizing the game goal, is constructed based on the Filippov-Castaing theorem on measurable choice. To illustrate the developed scheme, we analyze in detail the problem of approaching two controlled oscillation systems in geometric coordinates.