This article investigates a nonlinear approach-evasion differential game on a finite time interval. In the position space, the target set and the set that forms the state constraints by its time sections are given. It is assumed that the target set is closed in the position space with a coordinate-wise convergence topology, while the set generating the state constraints has closed time sections. With respect to the controlled system, the conditions of continuity of the right-hand side of the differential equation, uniqueness and uniform boundedness of the trajectories generated by relaxed controls (control measures) are assumed to be satisfied. Under these conditions, a theorem on the alternative solvability of the differential game is established in classes of positional control procedures: control scheme with a guide for a player interested in approaching the target set, and triplet-strategies including non-anticipating control of correction moments for an evading player. For the sets of successful solvability of players that define an alternative partition of the position space, representations are given in terms of the program iteration method constructions. The research goes back to the fundamental theorem on the alternative by N. N. Krasovskii and A. I. Subbotin and its subsequent extension to the class of systems with non-Lipschitz right-hand side (in the sense of state dependence), established by A. V. Kryazhimskii (in the aforementioned studies both the target set and the set forming the phase constraints supposed to be closed in the topology of coordinate-wise convergence).
Keywords: alternative, non-anticipative strategies, program iteration method, generalized program controls, guided control
Received February 10, 2026
Revised March 5, 2026
Accepted March 10, 2026
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 075-02-2026-737).
Aleksandr Georgievich Chentsov, Dr. Phys.-Math. Sci., Prof., Corresponding Member of RAS, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia; Ural Federal University Yekaterinburg, 620000 Russia, e-mail: chentsov@imm.uran.ru
Dmitrii Aleksandrovich Serkov, Dr. Phys.-Math. Sci., Prof. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 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. On alternative solvability in the approach-evasion problem. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 271–295.
