The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations

In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< \alpha \leq 1$ and the Riemann- Liouville fractional integral of order $\beta >0$ under certain constraints. A fractional analogue of the Du Bois-Reymond lemma is proved. Using this lemma for various weak local minimum problems, the Euler-Lagrange equation is derived in integral form. Some serious works in the literature claim that the standard proof of the Legendre condition in the classical case $\alpha=1$ cannot be adapted to the fractional case $0<\alpha <1$ with final constraints. In spite of this, we prove the Legendre conditions using the standard classical method. The obtained necessary conditions are illustrated by appropriate examples.