Neumann boundary geometric control of a fractional diffusion process

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.