A nonlinear conflict-controlled process involving two players is considered under a phase constraint. The process is viewed from the perspective of the first player. Its goal is to bring the controlled object to the terminal set under admissible control by the second player and the phase constraint is satisfied. It is assumed that the first player knows the dynamic characteristics of the controlled object, the phase variables, and the second player’s control up to the current moment. Sufficient conditions are presented under which the first player can guarantee the game’s phase vector will reach the terminal set under the phase constraints. A form of positional countercontrol by the first player that guarantees the completion of the convergence process is presented. Examples are provided.
Keywords: differential games, positional control, countercontrol, positional countercontrol, phase constraint, approach problem, structural synthesis, guided control, analytic design of aggregated controllers
Received May 13, 2026
Revised June 18, 2026
Accepted June 22, 2026
Funding Agency: The work was supported by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the Agreement No. 075-15-2025-345.
Nikolay Leontievich Grigorenko, Dr. Phis.-Math Sci., Prof., Faculty of Computational Mathematics and Cybernetics of the Lomonosov Moscow State University, Moscow, 119991 Russia; Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia, e-mail: grigor@cs.msu.ru
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Cite this article as: N.L. Grigorenko. On the problem of approach under phase constraint. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 44–56.
