Mathematics of Control, Signals and Systems

, Volume 18, Issue 3, pp 260–271

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

Original Article

DOI: 10.1007/s00498-006-0003-3

Cite this article as:
Miller, L. Math. Control Signals Syst. (2006) 18: 260. doi:10.1007/s00498-006-0003-3


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.


Interior controllabilitySpectral observabilityControl costParabolic equationFractional calculus

Copyright information

© Springer-Verlag London Limited 2006

Authors and Affiliations

  1. 1.Équipe Modal’X, EA 3454Université Paris XNanterreFrance
  2. 2.Centre de Mathématiques Laurent Schwartz, UMR CNRS 7640École PolytechniquePalaiseauFrance