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 |
Keywords:
Interior controllability; Spectral observability; Control cost; Parabolic equation; Fractional calculusReferences:
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