Section: New Results

Analysis, control and stabilization of heterogeneous systems

Motivated by the collision problem for rigid bodies in a perfect fluid, Munnier and Ramdani investigated in [9] the asymptotics of a 2D Laplace Neumann problem in a domain with cusp. The small parameter involved in the problem is the distance between the solid and the cavity's bottom. Denoting by $\alpha >0$ the tangency exponent at the contact point, the authors prove that the solid always reaches the cavity in finite time, but with a non zero velocity for $\alpha <2$ (real shock case), and with null velocity for $\alpha ⩾2$ (smooth landing case). The proof is based on a suitable change of variables transforming the Laplace Neumann problem into a generalized Neumann problem set on a domain containing a horizontal rectangle whose length tends to infinity as the solid approached the cavity.

The paper [14] presents the first positive result on approximate controllability for bilinear Schrödinger equations in presence of mixed spectrum when the interaction term is unbounded.

In [15] , Tucsnak, Valein and Wu study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems. The main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge to the optimal time and to the optimal controls of the original problem. This is based on a nonsmooth data error estimate for abstract parabolic systems.

A vesicle is an elastic membrane containing a liquid and surrounded by another liquid. Such a vesicle can be found in nature or it can be created in laboratory. They can store and/or transport substances. Modeling vesicles is also a first step in order to study and understand the behavior of more complex cells such as red cells. Their studies are important for many applications, in particular in biological and physiological subjects. Recent papers have been devoted to both experimental studies to the modeling and finally to the mathematical analysis of the obtained models. There are many different models to describe the motion of the membrane and one can for instance optimize the shape in order to minimize the elastic energy of the membrane. Such a problem is tackled in [4] in the 2D case and in [6] in the 3D case. In [4] , the optimization is done among convex domains whereas in [6] , the authors consider the problem of minimizing the total mean curvature in order to understand the differences between the Helfrich energy and the Willmore energy. Up to now, these models are considered without any fluid.

In [13] , San Martin, Takahashi and Tucsnak consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. They obtain the exact controllability of the prolate spheroidal swimmer and the existence of an optimal control problem (in the sense of the efficiency during a stroke). They also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. They analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, they use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid.