# American Institute of Mathematical Sciences

February  2018, 11(1): 1-19. doi: 10.3934/dcdss.2018001

## Optimality conditions for fractional variational problems with free terminal time

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

Received  June 2016 Revised  January 2017 Published  January 2018

Fund Project: 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

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.

Citation: Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001
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