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

Camille Laurent (1, 2) , Matthieu Léautaud (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CNRS - Centre National de la Recherche Scientifique
3 IMJ - Institut de Mathématiques de Jussieu

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 Hörmander. 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 Schrödinger equation.

Domaines: Equations aux dérivées partielles [math.AP]

Camille Laurent, Matthieu Léautaud. Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves.. 2015. ⟨hal-01163510⟩