2004
DOI: 10.1016/j.jde.2004.05.007
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Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Abstract: Earlier version on arXiv:math.AP/0307158Given a control region $\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $\Omega$. It is known that any initial data in $L^{2}(M)$ can be steered to $0$ in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^{2}([0,T]\times\Omega)$, 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 $\Omega$. %% 72 w… Show more

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Cited by 81 publications
(122 citation statements)
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“…This upper bound for the fast controllability cost in the case α = 1 was already stated without proof in [Mil04]. The first motivation for writing the proof came from the preprint version of [MZ04].…”
Section: Application To the Fractional Diffusionmentioning
confidence: 76%
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“…This upper bound for the fast controllability cost in the case α = 1 was already stated without proof in [Mil04]. The first motivation for writing the proof came from the preprint version of [MZ04].…”
Section: Application To the Fractional Diffusionmentioning
confidence: 76%
“…The fast null-controllability for any control region Ω has been known for a decade and the fast controllability cost has been investigated, e.g. [FCZ00,Mil04]. It allows us to discuss the optimality of the upper bound in th.2.1.…”
Section: Application To the Fractional Diffusionmentioning
confidence: 99%
See 3 more Smart Citations