The game problem of convergence of motions is considered for an abstract dynamical system with a given target set inside the phase constraints. An arbitrary subset of real numbers acts as a time "interval". The target set $\mathcal M$ and the phase constraints $\mathcal N$ obey the $\mathcal M\subset\mathcal N$ embedding. Nonanticipating multifunctions defined on the histories of disturbances are considered as admissible control strategies. A description of the solvability set and the construction of resolving control strategies based on the method of program iterations are given. At the same time, by increasing the "number"' of iterations of the program absorption operator, it is possible to expand (compared to the original version of the method) the areas of applicability due to the weakening or complete rejection of the topological requirements to the system dynamics, the target set, and phase constraints. The proposed constructions and their justification use the technique of fixed points of monotone mappings in partially ordered sets.
Keywords: convergence game problem, program iterations, abstract dynamical system, non-anticipating strategies
Received June 1, 2022
Revised July 11, 2022
Accepted July 18, 2022
Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-874).
Dmitrii Aleksandrovich Serkov, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: serkov@imm.uran.ru
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Cite this article as: D.A. Serkov. Transfinite version of the program iteration method in a convergence game problem for an abstract dynamical system. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 176–187; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S218–S228.
