Direct Method for Solving Game Problems of Approach of Controlled Objects

• Published in: Cybernetics and systems analysis Vol. 59; no. 5; pp. 812 - 820
• Main Authors: Chikrii, A. A., Rappoport, I. S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2023; Springer; Springer Nature B.V
• Subjects: Artificial Intelligence; Control; Differential games; Game theory; Mathematics; Mathematics and Statistics; Pontryagin principle; Processor Architectures; Software Engineering/Programming and Operating Systems; Systems Theory; resolving function; multi-valued mapping; stroboscopic strategy; quasilinear differential game; measurable selector
• ISSN: 1060-0396; 1573-8337
• DOI: 10.1007/s10559-023-00617-8

The authors analyze the problem of approach of controlled objects in dynamic game problems. The sufficient conditions for the game termination in a finite guaranteed time are obtained when the classical Pontryagin condition is not satisfied. Instead of the Pontryagin selection, which does not exist, some shift function is considered. With its help, a special set-valued mapping is introduced, which generates a lower resolving function. The latter plays a key role in formulating the result and allows constructing the control based on theorems of the Filippov–Castaing kind. A modified scheme of Pontryagin’s first direct method is proposed, which ensures a successful completion of the conflict-controlled process in the class of counter-controls. To compare the guaranteed times, the upper resolving function is introduced, and the corresponding scheme of the method is provided. The theoretical results are illustrated with a model example.