Section: New Results

Control and stabilization of heterogeneous systems

Participants : Thomas Chambrion, David Dos Santos Ferreira, Takéo Takahashi, Julie Valein.

• In [8], we find, thanks to a a semiclassical approach, $L^p$ estimates for the resolvants of the damped wave operator given on compact manifolds whose dimension is greater than 2.

• In [27], we have proved a “Ball-Marsedn-Slemrod” obstruction to the bi-linear controllability of the Klein-Gordon equation. With different methods, we obtained comparable results for the Gross-Pitaevskii equation in [28].

• In [7], we study the local exponential stability of the nonlinear Korteweg-de Vries equation with boundary time-delay feedback by using two different methods: a Lyapunov functional approach (with an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain) and an observability inequality approach (for any non critical lengths).

• In [12], we study the local controllability to trajectories of a Burgers equation with nonlocal viscosity. By linearization we are led to an equation with a non local term whose controllability properties are analyzed by using Fourier decomposition and biorthogonal techniques. Once the existence of controls is proved and the dependence of their norms with respect to the time is established for the linearized model, a fixed point method allows us to deduce the result for the nonlinear initial problem.

• In [26], we establish a Lebeau-Robbiano spectral inequality for a degenerated one dimensional elliptic operator and show how it can be used to impulse control and finite time stabilization for a degenerated parabolic equation.

• In [25], We prove a Carleman estimate in a neighborhood of a multi-interface, under compatibility assumptions between the Carleman weight, the operators at the multi-interface, and the elliptic operators in the interior and the usual sub-ellipticity condition. We derive some properties of unique prolongation, control of the heat equation, and satblization of the related damped waves equation.