Variational Problems Involving a Caputo-Type Fractional Derivative

TL;DR

This paper investigates calculus of variations problems involving a Caputo-type fractional derivative, providing necessary and sufficient conditions, and considering constraints, thus extending fractional calculus applications.

Contribution

It introduces new variational problems with a generalized Caputo-type fractional derivative and derives first and second order optimality conditions, including constraint cases.

Findings

Derived first and second order optimality conditions.

Extended calculus of variations to Caputo-type fractional derivatives.

Analyzed problems with integral and holonomic constraints.

Abstract

The aim of this paper is to study certain problems of calculus of variations, that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo--Hadamard fractional derivatives, that are dependent on a real parameter ro. Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered.