Variational integrator for fractional Pontryagin's systems. Existence of a discrete fractional Noether's theorem

TL;DR

This paper develops a variational integrator for fractional Pontryagin's systems to preserve their variational structure at the discrete level, introduces a fractional Noether's theorem, and tests convergence with an example.

Contribution

It presents a novel variational integrator for fractional Pontryagin's systems and establishes a discrete fractional Noether's theorem for systems with symmetry.

Findings

The integrator preserves the variational structure of fractional systems.

A convergence test confirms the effectiveness of the integrator.

The discrete Noether's theorem provides explicit constants of motion.

Abstract

Fractional Pontryagin's systems emerge in the study of a class of fractional optimal control problems but they are not resolvable in most cases. In this paper, we suggest a numerical approach for these fractional systems. Precisely, we construct a variational integrator allowing to preserve at the discrete level their intrinsic variational structure. The variational integrator obtained is then called shifted discrete fractional Pontryagin's system. We provide a solved fractional example in a certain sense. It allows us to test in this paper the convergence of the variational integrator constructed. Finally, we also provide a discrete fractional Noether's theorem giving the existence of an explicit computable discrete constant of motion for shifted discrete fractional Pontryagin's systems admitting a discrete symmetry.