Optimality conditions for fractional variational problems with free terminal time

Ricardo Almeida

doi:10.3934/dcdss.2018001

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2018, Volume 11, Issue 1: 1-19. Doi: 10.3934/dcdss.2018001

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Optimality conditions for fractional variational problems with free terminal time

Ricardo Almeida

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received: May 31, 2016

Revised: December 31, 2016

Published: February 2018

Work supported by Portuguese funds through the CIDMA -Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013. The author is grateful to two Reviewers, for several pertinent remarks, which improved the final version of the manuscript.

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Abstract

This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler-Lagrange equations are established for the fundamental problem and when in presence of an integral constraint. A Legendre condition, which is a second-order necessary condition, is also obtained. Other cases, such as the infinite horizon problem, the problem with delays in the Lagrangian, and the problem with high-order derivatives, are considered. Finally, a necessary condition for the optimal fractional order to satisfy is proved.

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Mathematics Subject Classification: Primary: 26A33, 49K05; Secondary: 34A08.

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