Original Article

Mathematics of Control, Signals and Systems

, Volume 18, Issue 3, pp 260-271

First online:

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

  • Luc MillerAffiliated withÉquipe Modal’X, EA 3454, Université Paris XCentre de Mathématiques Laurent Schwartz, UMR CNRS 7640, École Polytechnique Email author 

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Abstract

This paper introduces a “spectral observability condition” for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates, and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one-dimensional output systems from Müntz–Szász theorem on the closed span of sets of power functions.

Keywords

Interior controllability Spectral observability Control cost Parabolic equation Fractional calculus