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 a spectral inequality, i.e. 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_0 exp(c/T), or to the heat diffusion in potential wells observed from cones, i.e. c_0 exp(c/T^beta) with optimal beta. It also yields simple upper bounds for the cost rate c in terms of the spectral rate. This paper also gives geometric lower bounds on the spectral and cost rates for heat, diffusion and Ginzburg-Landau semigroups, including on non-compact Riemannian manifolds, based on L^2 Gaussian estimates.

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. Discrete and Continuous Dynamical Systems - Series B, 2010, 14 (4), pp.1465 - 1485. ⟨10.3934/dcdsb.2010.14.1465⟩. ⟨hal-00411846v3⟩