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