Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

Abstract

We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.

Keywords:

• dynamic optimization
• fractional calculus
• variable fractional order
• fractional calculus of variations

AMS Subject Classifications:

• 26A33
• 34A08
• 49K05

Acknowledgements

This work is part of first author’s PhD, which is carried out at the University of Aveiro under the Doctoral Programme Mathematics and Applications of Universities of Aveiro and Minho. The authors are grateful to three referees for valuable remarks and comments, which significantly contributed to the quality of the paper.

Notes

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA), and The Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013. Tavares was also supported by FCT through the PhD fellowship SFRH/BD/42557/2007; Torres by EU funding under the 7th Framework Programme FP7-PEOPLE-2010-ITN, [grant number 264735-SADCO], and by project PTDC/EEI-AUT/1450/2012, co-financed by FEDER under POFC-QREN with COMPETE reference FCOMP-01-0124-FEDER-028894.