On the controllability of anomalous diffusions generated by the fractional Laplacian

Luc Miller (1, 2)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 MODAL'X - Modélisation aléatoire de Paris X

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-Szasz theorem on the closed span of sets of power functions.

Mots clés: interior controllability; spectral observability; control cost; parabolic equation; fractional calculus

Domaines: Optimisation et contrôle [math.OC]; Analyse fonctionnelle [math.FA]

Luc Miller. On the controllability of anomalous diffusions generated by the fractional Laplacian. Mathematics of Control, Signals, and Systems, 2006, 18 (3), pp.260-271. ⟨10.1007/s00498-006-0003-3⟩. ⟨hal-00008809v4⟩