A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups

Luc Miller (1)

1 MODAL'X - Modélisation aléatoire de Paris X

This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces final-observability of an evolution semigroup from some stationary observability property on some spaces associated to the generator, e.g. spectral subspaces when the semigroup has an integral representation via spectral measures. Contrary to the original Lebeau-Robbiano strategy, it does not have recourse to null-controllability and it yields the optimal bound of the cost when applied to the heat equation, i.e. c' exp(c/T), or to the heat diffusion in potential wells observed from cones, i.e. c' exp(c/T^beta) with optimal beta. It also yields simple upper-bounds for c. This paper also gives a geometric lower bound on the rate in the estimate on sums of eigenfunctions of the Laplacian on which the original Lebeau-Robbiano strategy was based. Jerison and Lebeau had proved that this rate is positive.

Mots clés: Observability; null-controllability; fast control cost; heat semigroup

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Luc Miller. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. 2009. ⟨hal-00411846v2⟩