The abstract maximum principle and its application in the differential games theory

Vedyakov Alexei A., Anastasia O. Vedyakova, Slita Olga V., Tertychny-Dauri Vladimir Yu.

2026 , VOLUME 26, NUMBER 3 ( may-june )

ISSN 2226-1494 (print), ISSN 2500-0373 (online)

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Nikiforov
Vladimir O.
D.Sc., Prof.

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doi: 10.17586/2226-1494-2026-26-3-662-670

The abstract maximum principle and its application in the differential games theory

A. A. Vedyakov, A. O. Vedyakova, O. V. Slita, V. Y. Tertychny-Dauri

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Vedyakov A.A., Vedyakova A.O., Slita O.V., Tertychny-Dauri V.Yu. The abstract maximum principle and its application in the differential games theory. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2026, vol. 26, no. 3, pp. 662–670 (in Russian). doi: 10.17586/2226-1494-2026-26-3-662-670

Abstract

The problem of optimal control involving two opposing players is considered where optimality is understood in the minimax sense of achieving the best guaranteed outcome, and the control strategy is constructed with respect to the worst case admissible under the available measurements. The differential game problem is reduced to an optimal control synthesis problem by means of an abstract maximum principle using the method of Lagrange multipliers. A procedure is presented for applying the abstract maximum principle to the maximin formulation of the differential game problem within the Bellman framework in terms of dynamic programming. It is shown how the abstract maximum principle leads to the fundamental relations of Bellman’s optimization method for the differential game under consideration. The developed methodology for deriving optimality conditions in an antagonistic differential game using the abstract maximum principle can be applied to the analysis and design of nonlinear controlled dynamical systems with internally conflicting objectives.

Keywords: dynamic system, functional properties, Lagrange multipliers, maximum principle, differential game, optimal strategy

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