Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves

In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru [Tat95, Tat99b], Robbiano-Zuily [RZ98] and Hormander [Hor97]. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify those results to the wave operator on a Riemannian manifold M with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary partial derivative M. This allows us to obtain a global stability estimate from any open subset Gamma of M or partial derivative M, with the optimal time and dependence on the observation. As a first application, we compute a sharp lower estimate of the intensity of waves in the shadow of an obstacle. We also provide the cost of approximate controllability on the compact manifold M: for any T > 2 sup(x is an element of M) dist(x, Gamma), we can drive any H-0(1) x L-2 data in time T to an epsilon-neighborhood of zero in L-2 x H-1, with a control located in Gamma, at cost e(C/epsilon). We finally obtain related results for the Schrodinger equation.