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On the controllability of anomalous diffusions generated by the fractional Laplacian. (English) Zbl 1105.93015
Summary: 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.

93B05 Controllability
93C10 Nonlinear systems in control theory
93B07 Observability
26A33 Fractional derivatives and integrals
35B37 PDE in connection with control problems (MSC2000)
93B28 Operator-theoretic methods
Full Text: DOI
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