Section: New Results

Analysis and control of fluids and of fluid-structure interactions

In [47] , we analyze the system fluid-rigid body in the case of where the rigid body is a ball of “small radius”. More precisely, we consider the limit system as the radius goes to zero. We recover the Navier-Stokes system with a particle following the the velocity of the fluid. We consider in [45] a model of vesicle moving into a viscous incompressible fluid. Such a model, based on a phase-field approach was derived by researchers in Physics, and is quite difficult to study. By considering some approximation, we prove some result of existence of solutions for such a system.

By acting on a part of the fluid domain or on a part of the exterior boundary, we aim at controlling the fluid velocity, the rigid velocity and the position of the rigid body. It can be a control in open loop or in closed loop. We have studied both problems in the 1D case. In this case, the study benefits some simplifications, but can also be more difficult since the fluid domain is no more connected. As a consequence, if one wants to control by using only one input, on one part of the fluid domain, the fluid on the other side of the particle is only controlled by the motion of the structure.

We introduce a new method for controllability of nonlinear parabolic system allowing to deal with this problem and we solve it in ([24] ). We also obtain the local stabilization of such system around a stationary state in [41] .

We study the Cauchy problem corresponding to a similar 1D system without viscosity in [40] . In that case, we have to deal with the interaction between the particle and shock waves or relaxation waves. In [44] , we analyze a numerical scheme for the method of observers used to reconstruct the initial data of hyperbolic systems such as wave equation. We add some numerical viscosity in the scheme in order to have a uniform decay of the error between the reconstructed solution and the real one.

In [30] , a Lagrange-Galerkin method is introduced to approximate a two dimensional fluid-structure interaction problem for deformable solids. The new numerical scheme we present is based on a characteristics function mapping the approximate deformable body at the discrete time level $t_{k+1}$ into the approximate body at time $t_k$.

The aim of [25] is to tackle the time optimal controllability of an $(n+1)$-dimensional nonholonomic integrator with state constraints. A full description of an optimal control together with the corresponding optimal trajectories are explicitly obtained. The optimal trajectories we construct, are composed of arcs of circle lying in a 2-dimensional plane.

In [26] , controllability results are obtained for a low Reynolds number swimmer composed by a spherical object which is undergoing radial and axi-symmetric deformations in order to propel itself in a viscous fluid governed by the Stokes equations. A time optimal control problem is also solved for a simplified model and explicit optimal solutions are constructed.