2015/06/13 by Camille Laurent, Laurent, Camille, Matthieu Léautaud +1 · 3 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1506.04254
openalex publication_date 2015/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we first prove quantitative estimates associated to the\nunique continuation theorems for operators with partially analytic coefficients\nof Tataru, Robbiano-Zuily and H "ormander. We provide local stability estimates\nthat can be propagated, leading to global ones.\n Then, we specify the previous results to the wave operator on a Riemannian\nmanifold \M with boundary. For this operator, we also prove Carleman\nestimates and local quantitative unique continuation from and up to the\nboundary \∂ \M. This allows us to obtain a global stability\nestimate from any open set \Γ of \M or \∂ \M,\nwith the optimal time and dependence on the observation.\n This provides the cost of approximate controllability: for any T>2 \supx\n\∈ \M(dist(x,\Γ)), we can drive any data of H10 \× L2\nin time T to an \ε-neighborhood of zero in L2 \× H-1,\nwith a control located in \Γ, at cost eC/\ε.\n We also obtain similar results for the Schr "odinger equation.\n