For conflict-controlled dynamical systems satisfying the conditions of generalized uniqueness and uniform boundedness, the solvability of the minimax problem in the class of generalized controls is studied. The issues of consistency of such an extension are considered; i. e., the possibility of approximating generalized controls in the space of strategic measures by embeddings of ordinary controls is analyzed. For this purpose, the dependence of the set of measures on the general marginal distribution specified on one of the factors of the base space is studied. The continuity of this dependence in the Hausdorff metric defined by the metric corresponding to the *-weak topology in the space of measures is established. The density of embeddings of ordinary controls and control-noise pairs in sets of corresponding generalized controls in the *-weak topologies is also shown.
Keywords: generalized controls, strategic measures, minimax problem, *-weak convergence, Hausdorff metric
Received March 11, 2024
Revised March 27, 2024
Accepted April 1, 2024
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2024-1377).
Aleksandr Georgievich Chentsov, Dr. Phys.-Math. Sci., Prof., Corresponding Member RAS, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University Yekaterinburg, 620000 Russia, e-mail: chentsov@imm.uran.ru
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; Ural Federal University Yekaterinburg, 620000 Russia, e-mail: serkov@imm.uran.ru
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Cite this article as: A.G. Chentsov, D.A. Serkov. Continuous dependence of sets in a space of measures and a program minimax problem. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 2, pp. 277–299. Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325, Suppl. 1, pp. S76–S98.
