We consider a differential game in which a motion of a dynamical system is described by a functional-differential equation of retarded type. The goal of control of the first (respectively, second) player is to minimize (respectively, maximize) a cost functional consisting of two terms. The first term estimates the motion of the system and is specified by a functional that is defined on the space of continuous functions, is lower semicontinuous and locally bounded from above. The second term is an integral estimate of the system motion and controls of the players. Relaxing the standard assumption of continuity of the cost functional to semicontinuity allows to cover, in particular, some typical formulations of approach–evasion problems and problems on the minimax–maximin of the time to encounter with a given target set. We prove a theorem on the existence of the value of the differential game under consideration in classes of feedback (positional) players’ control strategies with memory of motion history. The proof is based on results from the theory of minimax (generalized) solutions of the corresponding to this game Cauchy problem for a path-dependent Hamilton–Jacobi equation with coinvariant derivatives under the right-end boundary condition and an appropriate variant of the extremal aiming method. As a corollary, and under an additional assumption of convexity of the (extended) vectogram of system velocities, we prove a theorem that the considered differential game, formalized in classes of players’ non-anticipative control strategies, also has the value, which coincides with the value of the game in classes of positional strategies. To prove this fact, a result on the existence of non-anticipative selections for non-anticipative multivalued maps is used.
Keywords: differential game, time-delay system, semicontinuous cost functional, feedback strategies, non-anticipative strategies, game value, path-dependent Hamilton–Jacobi equation, coinvariant derivatives, minimax solution
Received September 15, 2025
Revised November 3, 2025
Accepted November 10, 2025
Mikhail Igorevich Gomoyunov, Dr. Phys.-Math. Sci., 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: m.i.gomoyunov@gmail.com
Nikolai Yur’evich Lukoyanov, Dr. Phys.-Math. Sci., 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: nyul@imm.uran.ru
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Cite this article as: M.I. Gomoyunov, N.Yu. Lukoyanov. Existence of the value of a differential game for a time-delay system in the case of a semicontinuous cost functional. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 58–76.
