Section: New Results

Analysis and control of fluids and of fluid-structure interactions

In [42] , we consider a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. we investigate the asymptotic behavior of the Dirichlet energy associated to the solution of a Laplace Neumann problem as the distance between the solid and the cavity's bottom tends to zero. We prove that the solid always reaches the cavity in finite time. The contact occurs with non zero (real shock) or null velocity velocity (smooth landing), depending on the tangency exponent at the contact point. The proof is based on a suitable change of variables sending to infinity the cusp singularity at the contact. More precisely, the initial Laplace Neumann problem is transformed into a generalized Neumann problem set on a domain containing a horizontal strip, whose length goes to infinity as the the solid gets closer to the the cavity's bottom.

In [43] , we investigate the geometric inverse problem of determining, from the knowledge of the DtN operator of the problem, the positions and the velocities of moving rigid solids in a bounded cavity filled with a perfect fluid. We assume that the solids are small disks moving slowly. Using an integral formulation, we first derive the asymptotic expansion of the DtN map as the diameters of the disks tend to zero. Then, combining a suitable choice of exponential type data and the DORT technique (which is usually used in inverse scattering for the detection of point-like scatterers), we propose a reconstruction method for the unknown positions and velocities.

In [22] , Ana Leonor Silvestre (Lisbon, Portugal) and Takéo Takahashi analyze the system fluid-rigid body in the case of where the rigid body is a ball of “small radius”. More precisely, they consider the limit system as the radius goes to zero. They recover the Navier-Stokes system with a particle following the the velocity of the fluid.

In [14] , Mehdi Badra (University of Pau) and Takéo Takahashi study the feedback stabilization of a system composed by an incompressible viscous fluid and a rigid body. They stabilize the position and the velocity of the rigid body and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. The first result concerns weak solutions in the two-dimensional case, for initial data close to the stationary state. The method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. This additional difficulty leads to the assumption that the initial position of the rigid body is the position associated to the stationary state. Without this hypothesis, they work with strong solutions, and to deal with compatibility conditions at the initial time, they use finite dimensional dynamical controls. They prove again that for initial data close to the stationary state, they can stabilize the position and the velocity of the rigid body and the velocity of the fluid.

In [15] , Mehdi Badra (University of Pau) and Takéo Takahashi use the Fattorini criterion (more known as the Hautus criterion) to obtain the feedback stabilizability of general linear and nonlinear parabolic systems. They then consider flow systems described by coupled Navier-Stokes type equations (such as MHD system or micropolar fluid system) to obtain the stabilizability by only considering a unique continuation property of a stationary Stokes system.

In [36] , we use geometric control theory to investigate the existence and the design of optimal strokes for swimmers in Stokes of potential flows.