Fractional Noether's Theorem with Classical and Riemann-Liouville Derivatives

TL;DR

This paper extends Noether's theorem to fractional calculus of variations involving classical and Riemann-Liouville derivatives, deriving constants of motion for mixed classical/fractional extremals and applications to optimal control.

Contribution

It introduces a Noether symmetry theorem for fractional variational problems with classical and Riemann-Liouville derivatives, including Lagrangian, Hamiltonian, and optimal control cases.

Findings

Derived constants of motion for fractional extremals.

Established Noether's theorem for mixed classical/fractional problems.

Extended results to fractional optimal control scenarios.

Abstract

We prove a Noether type symmetry theorem to fractional problems of the calculus of variations with classical and Riemann-Liouville derivatives. As result, we obtain constants of motion (in the classical sense) that are valid along the mixed classical/fractional Euler-Lagrange extremals. Both Lagrangian and Hamiltonian versions of the Noether theorem are obtained. Finally, we extend our Noether's theorem to more general problems of optimal control with classical and Riemann-Liouville derivatives.