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On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

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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.

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Authors and Affiliations

  1. Équipe Modal’X, EA 3454, Université Paris X, Bât. G, 200 Av. de la République, 92001, Nanterre, France

    Luc Miller

  2. Centre de Mathématiques Laurent Schwartz, UMR CNRS 7640, École Polytechnique, 91128, Palaiseau, France

    Luc Miller

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  1. Luc Miller
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Correspondence to Luc Miller.

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Miller, L. On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian. Math. Control Signals Syst. 18, 260–271 (2006). https://doi.org/10.1007/s00498-006-0003-3

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  • DOI: https://doi.org/10.1007/s00498-006-0003-3

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