M.I. Gomoyunov. Extremal shift to accompanying points in a positional differential game for a fractional-order system ... P. 11-34

Vol. 25, no. 1, 2019

A two-person zero-sum differential game is considered. The motion of the dynamical system is described by an ordinary differential equation with a Caputo fractional derivative of order $\alpha\in(0,1)$. The performance index consists of two terms: the first depends on the motion of the system realized by the terminal time and the second includes an integral estimate of the realizations of the players' controls. The positional approach is applied to formalize the game in the "strategies - counter-strategies" and "counter-strategies - strategies" classes as well in the "strategies - strategies" class under the additional saddle point condition in the small game. In each case, the existence of the value and of the saddle point of the game is proved. The proofs are based on an appropriate modification of the method of extremal shift to accompanying points, which takes into account the specific properties of fractional-order systems.

Keywords: fractional-order differential equation, Caputo derivative, differential game, game value, positional strategy, counter-strategy, extremal shift

Received November 21, 2018

Revised January 20, 2019

Accepted January 21, 2019

Mikhail Igorevich Gomoyunov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: m.i.gomoyunov@gmail.com

Cite this article as:    M.I. Gomoyunov,  Extremal shift to accompanying points in a positional differential game for a fractional-order system, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 11–34 . 

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