On Nonstationary Problem of Motion Control in Conflict Situation

摘要

The mathematical theory of control under conflict and uncertainty provides a wide range of fundamental methods to study controlled dynamic processes of various nature. This paper considers the game problems of pursuit for nonstationary controlled processes of general type with a cylindrical terminal set. The investigation is closely related to L.S. Pontryagin first direct method and the method of resolving functions. The purpose of the paper is to derive sufficient conditions for the game termination for some guaranteed time in favor of the first player to provide the control implementing this result. In the development of the method of resolving functions the upper and lower resolving functions of two types are introduced in the form of support functions of special multivalued mappings. This made it possible to obtain conditions for the game termination in the class of quasi- and stroboscopic strategies. The comprehensive analysis of properties of special maltivalued mappings and their selectors allowed us to choose measurable controls by virtue of measurable choice theorem. A comparison of the guaranteed times of the above mentioned methods is given. In so doing the properties of L × B -measurability of the key multivalued mappings and the corresponding resolving functions − the support functions of these mappings are used. The property of superpositional measurability of above mentioned objects plays essential role in the method design. As a rule, in specific model examples the resolving functions are the large positive roots of certain quadratic equations that makes it possible to obtain solution in an analytical form.