Semi-classical observation sufficices for observability: wave and Schr\"odinger equations

For the wave and the Schr\"odinger equations we show how observability can be deduced from the observability of solutions localized in frequency according to a dyadic scale

Measure propagation along C0\mathscr{C}^0-vector field and wave controllability on a rough compact manifold

The celebrated Rauch-Taylor/Bardos-Lebeau-Rauch geometric control condition is central in the study of the observability of the wave equation linking this property to high-frequency propagation along geodesics that are therays of geometric optics. This connection is best understood through the propagation properties of microlocal defect measures that appear as solutions to the wave equation concentrate. For a sufficiently smooth metric this propagation occurs along the bicharacteristic flow. If one considers a merely $\mathscr{C}^1$-metric this bicharacteristic flow may however not exist. The Hamiltonian vector field is only continuous; bicharacteristics do exist (as integral curves of this continuous vector field) but uniqueness is lost. Here, on a compact manifold without boundary, we consider this low regularity setting, revisit the geometric control condition, and address the question of support propagation for a measure solution to an ODE with continuous coefficients. This leads to a sufficient condition for the observability and equivalently the exact controllability of the wave equation. Moreover, we investigate the stabililty of the observability property and the sensitivity of the control process under a perturbation of the metric of regularity as low as Lipschitz

Exact controllability for the Lamé system

International audienc

LpL^p Carleman estimates for elliptic boundary value problems and applications to the quantification of unique continuation

The aim of this work is to prove global $L^p$ Carleman estimates for the Laplace operator in dimension $d \geq 3$. Our strategy relies on precise Carleman estimates in strips, and a suitable gluing of local and boundary estimates obtained through a change of variables. The delicate point and most of the work thus consists in proving Carleman estimates in the strip with a linear weight function for a second order operator with coefficients depending linearly on the normal variable. This is done by constructing an explicit parametrix for the conjugated operator, which is estimated through the use of Stein Tomas restriction theorems. As an application, we deduce quantified versions of the unique continuation property for solutions of $\Delta u = V u + W_1 \cdot \nabla u + \div(W_2 u)$ in terms of the norms of $V$ in $L^{q_0}(\Omega)$, of $W_1$ in $L^{q_1}(\Omega)$ and of $W_2$ in $L^{q_2}(\Omega)$ for $q_0 \in (d/2, \infty]$ and $q_1$ and $q_2$ satisfying either $q_1, \, q_2 > (3d-2)/2$ and $1/q_1 + 1/q_2 3d/2$.Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locau

Semi-classical observation suffices for observability : wave and Schrödinger equations

For the wave and the Schrödinger equations we show how observability can be deduced from the observability of solutions localized in frequency according to a dyadic scale

Semi-Classical Observation Suffices for Observability: Wave and Schrödinger Equations

For the linear wave and Schrödinger equations we show how observability can be deduced from the observability of solutions localized in frequency with a dyadic scale

MEASURE PROPAGATION ALONG C^0-VECTOR FIELD AND WAVE CONTROLLABILITY ON A ROUGH COMPACT MANIFOLD

The celebrated Rauch-Taylor/Bardos-Lebeau-Rauch geometric control condition is central in the study of the observability of the wave equation linking this propery to highfrequency propagation along geodesics that are the rays of geometric optics. This connection is best understood through the propagation properties of microlocal defect measures that appear as solutions to the wave equation concentrate. For a sufficiently smooth metric this propagation occurs along the bicharacteristic flow. If one considers a merely C 1-metric this bicharacteristic flow may however not exist. The Hamiltonian vector field is only continuous; bicharacteristics do exist (as integral curves of this continuous vector field) but uniqueness is lost. Here, on a compact manifold without boundary, we consider this low regularity setting, revisit the geometric control condition, and address the question of support propagation for a measure solution to an ODE with continuous coefficients. This leads to a sufficient condition for the observability and equivalently the exact controllability of the wave equation. Moreover, we investigate the stabililty of the observability property and the sensitivity of the control process under a perturbation of the metric of regularity as low as Lipschitz