Solving the Problem of Approach of Controlled Objects in Dynamic Game Problems

The problem of a guaranteed result in game problems of approach of controlled objects is considered. A method for solving such problems is proposed. It involves constructing some scalar functions that qualitatively characterize the course of approach of controlled objects and the efficiency of decisions. Such functions are called resolving functions. In contrast to the main scheme of the method, the case is considered where the classical Pontryagin condition does not hold. In this situation, instead of the Pontryagin selector, which does not exist, some shift functions are considered and special multivalued mappings are introduced with their help. They generate upper and lower resolving functions, which are used to formulate the sufficient conditions for the game completion in a certain guaranteed time. An example is given to illustrate the approach of controlled objects with a simple motion, in order to obtain upper and lower resolving functions in explicit form, which allows making a conclusion about the possibility of ending the game when the Pontryagin condition does not hold.