# American Institute of Mathematical Sciences

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November  2010, 14(4): 1465-1485. doi: 10.3934/dcdsb.2010.14.1465

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

 1 MODAL’X, EA 3454, Bât. G, Université Paris Ouest Nanterre La Défense, 200 Av. de la République, 92001 Nanterre Cedex, France

Received  August 2009 Revised  February 2010 Published  August 2010

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.
Citation: Luc Miller. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1465-1485. doi: 10.3934/dcdsb.2010.14.1465
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