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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2019 Volume 25, Number 1, Pages 11–34 (Mi timm1597)

This article is cited in 12 papers

Extremal Shift to Accompanying Points in a Positional Differential Game for a Fractional-Order System

M. I. Gomoyunovab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: A two-person zero-sum differential game is considered. The motion of the dynamic system is described by an ordinary differential equation with a Caputo fractional derivative of order $\alpha\in(0,1)$. The quality 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 “strategy–counterstrategy” and “counterstrategy–strategy” classes as well as in the “strategy–strategy” classes 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, counterstrategy, extremal shift.

UDC: 517.977

MSC: 49N79, 34K37

Received: 22.11.2018
Revised: 20.01.2019
Accepted: 21.01.2019

DOI: 10.21538/0134-4889-2019-25-1-11-34


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2020, 308, suppl. 1, S83–S105

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