A direct Lebeau-Robbiano strategy for the observability of
Luc Miller - MODAL’X, EA 3454, Bât. G, Université Paris Ouest Nanterre La Défense, 200 Av. de la République, 92001 Nanterre Cedex, France (email)
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.
null-controllability, fast control cost, heat semigroup.
Received: August 2009; Revised: February 2010; Published: August 2010.
2011 Impact Factor.921