Fractional variational problems depending on indefinite integrals

TL;DR

This paper derives necessary optimality conditions for fractional variational problems involving Caputo derivatives and indefinite integrals, generalizing existing results and covering various constrained and higher-order cases.

Contribution

It introduces a comprehensive set of fractional Euler-Lagrange equations and boundary conditions for complex variational problems with indefinite integrals and constraints.

Findings

Derived fractional Euler-Lagrange equations for Caputo derivatives

Established natural boundary conditions for these problems

Extended results to isoperimetric and holonomic constrained problems

Abstract

We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered.