In this article, we first prove quantitative estimates associated to the unique continuation theorems for operators with partially analytic coefficients of Tataru, Robbiano-Zuily and H\"ormander. We provide local stability estimates that can be propagated, leading to global ones. Then, we specify the previous results to the wave operator on a Riemannian manifold $\mathcal{ M}$ with boundary. For this operator, we also prove Carleman estimates and local quantitative unique continuation from and up to the boundary $\partial \mathcal{ M}$. This allows us to obtain a global stability estimate from any open set $\Gamma$ of $\mathcal{ M}$ or $\partial \mathcal{ M}$, with the optimal time and dependence on the observation. This provides the cost of approximate controllability: for any $T>2 \sup_{ x \in \mathcal{ M}}(dist(x,\Gamma))$, we can drive any data of $H^1_0 \times L^2$ in time $T$ to an $\varepsilon$-neighborhood of zero in $L^2 \times H^{ -1}$, with a control located in $\Gamma$, at cost $e^{ C/\varepsilon}$. We also obtain similar results for the Schr\"odinger equation.
Categories: math.AP