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

摘要

In this paper, we study a class of fractional optimal control problems. Anecessary condition for the existence of an optimal control is provided in theliterature. It is commonly given as the existence of a solution of a fractionalPontryagin's system and the proof is based on the introduction of a Lagrangemultiplier. Assuming an additional condition on these problems, we suggest anew presentation of this result with a proof using only classical mathematicaltools adapted to the fractional case: calculus of variations, Gronwall's Lemma,Cauchy-Lipschitz Theorem and stability under perturbations of differentialequations. In this paper, we furthermore provide a way in order to transit froma classical optimal control problem to its fractional version via theStanislavsky's formalism. We also solve a strict fractional example allowing totest numerical schemes. Finally, we state a fractional Noether's theorem givingthe existence of an explicit constant of motion for fractional Pontryagin'ssystems admitting a symmetry.