Control from an Interior Hypersurface

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