Control From an Interior Hypersurface

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.