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

Pages: 1465 - 1485, Volume 14, Issue 4, November 2010

doi:10.3934/dcdsb.2010.14.1465       Abstract        Full Text (359.7K)       Related Articles

Luc Miller - MODAL’X, EA 3454, Bât. G, Université Paris Ouest Nanterre La Défense, 200 Av. de la République, 92001 Nanterre Cedex, France (email)

Abstract: 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. words 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.

Keywords:  observability, null-controllability, fast control cost, heat semigroup.
Mathematics Subject Classification:  93B07, 35B37.

Received: August 2009;      Revised: February 2010;      Published: August 2010.