Title: Neumann boundary geometric control of a fractional diffusion process

Authors: Ahmed Maidi; Jean-Pierre Corriou

Addresses: Laboratoire de Conception et Conduite des Systémes de Production, Université Mouloud MAMMERI, 15 000, Tizi-Ouzou, Algeria ' Laboratoire Réactions et Génie des Procédés, UMR 7274-CNRS, Lorraine Université, ENSIC 1, rue Grandville, BP 20451, 54001, Nancy Cedex, France

Abstract: In this paper, a boundary controller is developed for the fractional diffusion equation in the framework of geometric control. The case of Neumann actuation with a spatial weighted average output is addressed. This non-collocated configuration is characterised by an infinite characteristic index. To overcome this difficulty, the notion of the extended operator is exploited to derive an equivalent distributed control problem. The equivalent model, obtained by the Laplace transform in space domain, is used both for controller design and stability analysis of the resulting closed-loop. Thus, based on the notion of the characteristic index, a state feedback control that enforces an output tracking is designed. Then, the exponential stability of the closed-loop is demonstrated based on the semigroup theory. The effectiveness of the developed controller is shown, through numerical simulation, in the case of a heated aluminium rod that exhibits an anomalous diffusion phenomenon.

Keywords: distributed parameter system; spatial fractional partial differential equation; fractional diffusion equation; Neumann actuation; geometric control; characteristic index.

DOI: 10.1504/IJMIC.2020.108915

International Journal of Modelling, Identification and Control, 2020 Vol.34 No.1, pp.51 - 58

Received: 21 Jan 2020
Accepted: 22 Jan 2020

Published online: 06 Aug 2020 *

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