Optimality conditions for fractional variational problems with free terminal time

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.