A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups
Résumé
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 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' exp(c/T), or to the heat diffusion in potential wells observed from cones, i.e. c' exp(c/T^beta) with optimal beta. It also yields simple upper-bounds for c.
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