A class of fractional optimal control problems and fractional Pontryagin's systems. Existence of a fractional Noether's theorem

TL;DR

This paper investigates fractional optimal control problems, providing new existence results, a method to transition from classical to fractional problems, and establishing a fractional Noether's theorem with explicit constants of motion.

Contribution

It offers a novel proof of the existence of solutions using classical tools, introduces a way to convert classical problems to fractional ones, and proves a fractional Noether's theorem.

Findings

Established a new proof for the existence of solutions using classical mathematical tools.

Provided a method to transition from classical to fractional optimal control problems.

Proved a fractional Noether's theorem with explicit constants of motion.

Abstract

In this paper, we study a class of fractional optimal control problems. A necessary condition for the existence of an optimal control is provided in the literature. It is commonly given as the existence of a solution of a fractional Pontryagin's system and the proof is based on the introduction of a Lagrange multiplier. Assuming an additional condition on these problems, we suggest a new presentation of this result with a proof using only classical mathematical tools adapted to the fractional case: calculus of variations, Gronwall's Lemma, Cauchy-Lipschitz Theorem and stability under perturbations of differential equations. In this paper, we furthermore provide a way in order to transit from a classical optimal control problem to its fractional version via the Stanislavsky's formalism. We also solve a strict fractional example allowing to test numerical schemes. Finally, we state a fractional Noether's theorem giving the existence of an explicit constant of motion for fractional Pontryagin's systems admitting a symmetry.