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

Abstract

This paper generalizes and simplifies abstract results of Miller and Seidmanon the cost of fast control/observation. It deduces final-observability of an evolution semigroupfrom a spectral inequality, i.e. some stationary observability propertyon some spaces associated to the generator,e.g. spectral subspaces when the semigrouphas an integral representation via spectral measures. wordsContrary to the original Lebeau-Robbiano strategy,it does not have recourse to null-controllabilityand 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 ratesfor heat, diffusion and Ginzburg-Landau semigroups,including on non-compact Riemannian manifolds,based on $L^2$ Gaussian estimates.