# Optimality conditions for fractional variational problems with free terminal time

## TL;DR

This paper establishes necessary and sufficient optimality conditions for fractional variational problems involving fractional derivatives, including Euler-Lagrange equations, Legendre conditions, and extensions to various complex cases.

## Contribution

It introduces comprehensive optimality conditions for fractional variational problems with free terminal time, covering multiple advanced scenarios and fractional order considerations.

## Key findings

- Derived fractional Euler-Lagrange equations for fundamental and constrained problems
- Established a second-order Legendre condition for optimality
- Provided conditions for the optimal fractional order

## Abstract

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.