In this research project, we will investigate control and inverse problems of some linear and nonlinear partial dfferential equations (PDE). Let us be more precise about the equations and the problems we plan to address.

Controllability of the Zakharov system. We are interested in control and stabilization properties of the Zakharov system, which consists of a Schrödinger equation coupled with a wave equation. Since no control results are known for this system, we will start our study in a onedimensional setting by means of distributed controls and periodic boundary conditions. Then, we will investigate the feedback stabilization. For both problems, we will combine the methods known for the Schödinger and wave equations in order to obtain the desired results.

Control of the stabilized KuramotoSivashinsky system. We consider a nonlinear system of PDE where a KuramotoSivashinsky (KS) equation interacts with a heat equation, known as the stabilized KS system. The local null controllability of this system has been proven by means of boundary controls on the leftend of the interval. Recently, the local null controllability has been obtained with an internal control acting only on the KS equation. Here, we want to fill the gap and prove the same result by only controlling the heat equation with a distributed control. The main novelty will be a new Carleman estimate for a fourthorder parabolic equation with nonhomogeneous boundary conditions, which will be used to deal with the coupling of the equations that occurs at the first order.

On the uniform controllability of a linear Kortewegde Vries equation. We will study the behavior of the cost of null controllability of a linear Kortewegde Vries equation in the vanishing dispersion limit with mixed boundary conditions. When the dispersion coecient is not present, we find a transport equation which we know is null controllable for large times using controls equal to zero. Therefore, when a suciently large time has passed we expect that the norm of the control would tend to zero as the dispersion coecient does. The principal objective will be to prove an exponential dissipation estimate in order to compensate the observability constant coming from a Carleman inequality, which blows up as the dispersion coefficient vanishes.

Controllability to the trajectories of the NavierStokes system with controls having one vanishing component. The controllability of the Ndimensional NavierStokes system by controls having a reduced number of components has been the subject of study of several works. The first result concerning the controllability to the trajectories in this matter was obtained under the assumption that the control region touched the boundary of the domain. Recently, this hypothesis has been removed in the context of null controllability. The advantage is that the linearization around zero allows to decouple the equations satisfied by the components of the fluid. Here, we intend to prove the local null controllability to the trajectories with controls having one null component and no restriction on the control domain. The usual approach is to establish an appropriate Carleman estimate for the linearized system, that is, with local terms of only N 1 components of the solution. Special attention will be paid to the presence of the target trajectory on the linearized system, which does not allow the decoupling of the equations and, therefore, the previous methods difficult to apply.

Inverse problem for a linear elasticity system. We will be concerned with the recovery of the elasticity coecients and the density of a linear isotropic elasticity system from measurements of the displacement. In particular, we will be concerned with the uniqueness and stability aspects of this inverse problem. This problem has been treated with success in the past years. Here, we propose to improve these results by considering measurements only on a limited number of components of the displacement vector which can be made on a part of the boundary or in an interior subdomain. By following the BukhgeimKlibanov method, we plan to prove a Carleman estimate for this system where the local term does not contain the missing directions of the measurements.
