Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives

We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler–Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered.