Game Dynamic Problems for Systems with Fractional Derivatives

Чикрий, А. А.

doi:10.1007/978-0-387-77247-9_13

Cited by 19 publications

(6 citation statements)

References 17 publications

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“…We will use an example of classical Riemann-Liouville fractional derivatives [27,28], regularized fractional derivatives (independently introduced and used by Dzhrbashyan-Nersesyan and Caputo), Miller-Ross sequential fractional derivatives [68], and Hilfer fractional derivatives [67] to show how these derivatives are used to solve dynamic game problems. The early studies in this field were carried out by Chikrii and Eidel'man in [27,28,97,98] and Chikrii in [56,57], and now they are continued by Matichin [67][68][69][70]80].…”

Section: Processes With Fractional Derivatives: Game Problems Generamentioning

confidence: 99%

Developing B. N. Pshenichnyi’s scientific ideas in optimization and mathematical control theory

Сергиенко

Чикрий

2012

Cybern Syst Anal

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The paper overviews the studies initiated by fundamental works of B. N. Pshenichnyi in the field of necessary extremum conditions, numerical optimization methods, optimal control theory, and differential games. Among them are new scientific fields such as processes with fractional derivatives and impulsive control systems.This year marks the 75th anniversary of the birth of B. N. Pshenichnyi, Academician of the National Academy of Sciences (NAS) of Ukraine, a world famous Ukrainian scientist, an expert in mathematics and cybernetics.Boris Nikolaevich Pshenichnyi graduated from the faculty of mechanics and mathematics of the Ivan Franko National University of L'viv, and the most part of his life worked in Kyiv at the V. M. Glushkov Institute of Cybernetics of the NAS of Ukraine. The range of scientific interests of B. N. Pshenichnyi was extraordinary and included network design and graph 157 1060-0396/12/4802-0157

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Section: Processes With Fractional Derivatives: Game Problems Generamentioning

confidence: 99%

Developing B. N. Pshenichnyi’s scientific ideas in optimization and mathematical control theory

Сергиенко

Чикрий

2012

Cybern Syst Anal

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“…Models of differential games in which derivatives of non-integer orders are used and, thereby, the power-fading memory is taken into account were proposed in the works of Arkadiy A. Chikriy (Arkadii Chikrii), Ivan I. Matychyn and Alexander G. Chentsov [253][254][255][256][257] (see also [258][259][260]), which are clearly not related to economy. Note that the construction of models of economic behavior, using differential games with power memory, instead of games with full memory, currently remains an open question.…”

Section: Fractional Differential Games In Economicsmentioning

confidence: 99%

“…The construction of such models requires further research on economic behavior within the framework of game theory. The basis for such constructions can be the methods and results described in [253][254][255][256][257][258][259][260].…”

Section: Fractional Differential Games In Economicsmentioning

confidence: 99%

On History of Mathematical Economics: Application of Fractional Calculus

Tarasov

2019

Mathematics

182

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Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed.

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“…Our study is based on the approaches to the solution of game control problems from [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The approach to the solution of the described class of problems proposed in the present paper is based on analyzing the original problem by means of necessary optimality conditions under the assumption of complete information about the uncertainty on a control interval under consideration.…”

Section: Introductionmentioning

confidence: 99%

A control problem with dominating uncertainty

Grigorenko

2014

Proc. Steklov Inst. Math.

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A control problem for a nonstationary dynamic system with nonfixed termination time and terminal functional in the presence of uncertain parameters is considered. A control construction algorithm is proposed, which uses elements of necessary optimality conditions and the speed gradient principle providing a guaranteed value of the quality functional. The results of computing the control and values of the functional for test parameters of the model are presented.

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Game Dynamic Problems for Systems with Fractional Derivatives

Cited by 19 publications

References 17 publications

Developing B. N. Pshenichnyi’s scientific ideas in optimization and mathematical control theory

Developing B. N. Pshenichnyi’s scientific ideas in optimization and mathematical control theory

On History of Mathematical Economics: Application of Fractional Calculus

A control problem with dominating uncertainty

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