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

2012/03/08 by Loïc Bourdin, Bourdin, Loïc
Engineering · Mathematics · #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #math.OC

paper · pdf · doi:10.48550/arxiv.1203.1707

27 pages, 15 figures

arxiv created 2012/03/08 · openalex publication_date 2012/03/08 · arxiv updated 2012/03/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

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