Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time |
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Authors: | Luc Miller |
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Institution: | a Équipe Modal'X, EA 3454, Université Paris X, Bât. G, 200 Av. de la République, 92001 Nanterre, France b Centre de Mathématiques Laurent Schwartz, UMR CNRS 7640, École Polytechnique, 91128 Palaiseau, France |
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Abstract: | Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Ω. It is known that any initial data in L2(M) can be steered to 0 in an arbitrarily small time T by applying a suitable control g in L2(0,T]×Ω), and, as T tends to 0, the norm of g grows like exp(C/T) times the norm of the data. We investigate how C depends on the geometry of Ω. We prove C?d2/4 where d is the largest distance of a point in M from Ω. When M is a segment of length L controlled at one end, we prove for some . Moreover, this bound implies where is the length of the longest generalized geodesic in M which does not intersect Ω. The control transmutation method used in proving this last result is of a broader interest. |
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Keywords: | Heat equation Control cost Null-controllability Observability Small time asymptotics Multipliers Entire functions Transmutation |
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