The problem of a guaranteed result in game problems of group approach of controlled objects is considered. A method for solving such problems is proposed, which is associated with the construction of some scalar functions that qualitatively characterize the progress of the approach of a group of controlled objects and the efficiency of the decisions made. Such functions are called resolving functions. The attractiveness of the method of resolving functions lies in the fact that it makes it possible to use effectively the modern technique of multivalued mappings and their selection in substantiating game constructions and obtaining meaningful results on their basis. In any form of the method of resolving functions, the main principle is the accumulative principle, which is used in the current summation of the resolving functions to assess the quality of the game of the group approach until a certain threshold value is reached. In contrast to the main scheme of the mentioned method, the case is considered when the classical Pontryagin condition does not hold. In this situation, instead of Pontryagin’s selection, which do not exist, some shift functions are considered and, with their help, special multivalued mappings are introduced. They generate upper and lower resolving functions with the help of which sufficient conditions for the completion of the game of group approach in a certain guaranteed time are formulated. Comparison of guaranteed times for different schemes of group approach of controlled objects is given.
