Control from an interior hypersurface

Galkowski, Jeffrey; Léautaud, Matthieu

doi:10.1090/tran/7938

Control from an interior hypersurface
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by Jeffrey Galkowski and Matthieu Léautaud PDF

Trans. Amer. Math. Soc. 373 (2020), 3177-3233

Abstract:

We consider a compact Riemannian manifold $M$ (possibly with boundary) and $\Sigma \subset M\setminus \partial M$ an interior hypersurface (possibly with boundary). We study observation and control from $\Sigma$ for both the wave and heat equations. For the wave equation, we prove controllability from $\Sigma$ in time $T$ under the assumption $(\mathcal {T}$GCC) that all generalized bicharacteristics intersect $\Sigma$ transversally in the time interval $(0,T)$. For the heat equation we prove unconditional controllability from $\Sigma$. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on $\Sigma$ under $\mathcal {T}$GCC and unconditional exponential lower bounds on such Cauchy data.

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