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

TL;DR

This paper establishes necessary and sufficient optimality conditions for fractional calculus of variations problems involving Caputo derivatives, including Euler-Lagrange equations and isoperimetric problems, under convexity assumptions.

Contribution

It provides new optimality conditions and Euler-Lagrange equations for variational problems with Caputo derivatives, addressing cases with different bounds and integral constraints.

Findings

Derived Euler-Lagrange equations for Caputo derivatives with distinct bounds.

Established sufficient conditions for minimization under convexity.

Formulated fractional isoperimetric problems with Caputo derivatives.

Abstract

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.