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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.

##### MSC:
 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
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