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A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. (English) Zbl 1219.93017
Summary: 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 properties 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., $$c_0\exp(c/T)$$, 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 non-compact Riemannian manifolds, based on $$L^2$$ Gaussian estimates.

MSC:
 93B07 Observability 93B05 Controllability 35K90 Abstract parabolic equations 35Q93 PDEs in connection with control and optimization
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