Fractional conservation laws in optimal control theory

TL;DR

This paper extends Noether's theorem to fractional optimal control, revealing that conservation laws involve additional terms depending on fractional derivatives, thus generalizing classical results.

Contribution

It introduces a Noether-like theorem for fractional optimal control problems, showing how conservation laws are modified by fractional derivatives.

Findings

The fractional Hamiltonian does not define a conservation law in the fractional case.

A new fractional conservation law involves the Hamiltonian, generalized momentum, and fractional derivatives.

The results generalize classical conservation laws to fractional calculus contexts.

Abstract

Using the recent formulation of Noether's theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noether-like theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum, and the fractional derivative of the state variable.