Fractional variational problems depending on indefinite integrals and with delay

TL;DR

This paper establishes a necessary and sufficient condition for optimality in fractional variational problems that involve indefinite integrals and time delays, with an illustrative example demonstrating the analytical solution.

Contribution

It introduces a comprehensive optimality condition for fractional variational problems with delays and indefinite integrals, expanding the theoretical framework in this area.

Findings

Derived a necessary and sufficient optimality condition

Provided an analytical example of the minimizer

Extended fractional variational calculus to include delays

Abstract

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example, where we find analytically the minimizer.