# American Institute of Mathematical Sciences

February  2018, 11(1): 143-154. doi: 10.3934/dcdss.2018009

## Fractional Herglotz variational problems of variable order

 1 ESECS, Polytechnic Institute of Leiria, 2411–901 Leiria, Portugal 2 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal

* Corresponding author: Dina Tavares

Received  July 2016 Revised  February 2017 Published  January 2018

Fund Project: This work is part of first author's Ph.D., which is carried out at the University of Aveiro under the Doctoral Programme Mathematics and Applications of Universities of Aveiro and Minho. It was supported by Portuguese funds through CIDMA and The Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013. Tavares was also supported by FCT through the Ph.D. fellowship SFRH/BD/42557/2007

We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.

Citation: Dina Tavares, Ricardo Almeida, Delfim F. M. Torres. Fractional Herglotz variational problems of variable order. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 143-154. doi: 10.3934/dcdss.2018009
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##### References:
Graphics of function $z(\overline x, t)$
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