We consider a second-order hyperbolic equation in a bounded domain, for example the wave equation or the elasticity system. On this system we assume that the initial condition is unknown, but that we have at our disposal some measurements in time on the solution. Typically, for the wave equation we can assume that we measure (1) the derivative of the field (i.e. the "velocity") in a subdomain or on the boundary, or (2) directly the solution field on a boundary or in a subdomain. In each case we may have additional measurement noise on these observations, and therefore we assume to not be able to time-differentiate the measurements in the case (2). From this input, we derived a family of observers that allows to start from potentially wrong initial conditions and to use the data to converge over time to the measured solution. In the first case, the observer is inspired from the the stabilization strategy known as Direct Velocity Feedback (DVF) in the structural mechanics community. We have analysed the corresponding observer strategy in a rather general context, including cases where white noises are present in the data. In the case (2), we proposed an original observer that we call Schur Displacement Feedback (SDF) and that directly uses the measured field without time differentiation. We demonstrated its convergence based on an original observation inequality. This observer was extended in elasticity in situations where we only measure the shape of the deformed model - by medical imaging techniques for living organs - or when we access the deformation - typically of a mechanical system like the heart - through the new tagged-MRI imaging modalities.
Inria/École Polytechnique joint research team