## Section: New Results

### Control and stabilization of heterogeneous systems

#### Analysis of heterogeneous systems

Participants : Jean-François Scheid, Takéo Takahashi.

In [12], we consider a single disk moving under the influence of a 2D viscous fluid and study the asymptotic as the size of the solid tends to zero. If the density of the solid is independent of the size, the energy equality is not sufficient to obtain a uniform estimate for the solid velocity. This is achieved thanks to the optimal ${L}^{p}-{L}^{q}$ decay estimates of the semigroup associated to the fluid-rigid body system and to a fixed point argument.

In [10], we propose a new model for the motion of a viscous incompressible fluid. More precisely, we consider the Navier-Stokes system with a boundary condition governed by the Coulomb friction law. With this boundary condition, the fluid can slip on the boundary if the tangential component of the stress tensor is too large. We prove the existence and uniqueness of a weak solution in the two-dimensional problem and the existence of at least one solution in the three-dimensional case. In [9], we consider this model with a rigid body. We prove that there exists a weak solution for the corresponding system.

In [13], we study a free boundary problem modeling the motion of a piston in a viscous gas. The gas-piston system fills a cylinder with fixed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is modeled by the 1D compressible Navier-Stokes system and the piston motion is described by the second Newton's law. We prove the existence and uniqueness of global in time strong solutions. The main novelty is that we include the case of non homogeneous boundary conditions.

In [31], we study the shape differentiability of the free-boundary 1-dimensional simplified model for a fluid-elasticity system. The full characterization of the associated material derivatives is given and the shape derivative of an energy functional has been obtained.

#### Control of heterogeneous systems

Participants : Thomas Chambrion, Alessandro Duca, Takéo Takahashi.

In [11], we consider the swimming into a stationary Navier-Stokes fluid. The swimmer is a rigid body $\mathcal{S}\subset {\mathbb{R}}^{3}$ immersed in an infinitely extended fluid. We are interested in self-propelled motions of $\mathcal{S}$ in the steady state regime of the rigid body-fluid system, assuming that the mechanism used by the body to reach such a motion is modeled through a distribution of velocities on the boundary. We show that this can be solved as a control problem.

In [16] we prove that the Kuramoto-Sivashinsky equation is locally controllable in 1D and in 2D with one boundary control. His method consists in combining several general results in order to reduce the null-controllability of this nonlinear parabolic equation to the exact controllability of a linear beam or plate system. This improves known results on the controllability of Kuramoto-Sivashinsky equation and gives a general strategy to handle the null-controllability of nonlinear parabolic systems.

The paper [21] is the result of a long term analysis about the restrictions to the controllability of bilinear systems induced by the regularity of the propagators for the bilinear Schrödinger equation. This paper comes along with its companion paper [20] which gives a detailed proof of the celebrated Ball-Marsden-Slemrod obstruction to exact controllability for bilinear systems with ${L}^{1}$ controls.

The paper [23] is concerned with the one dimensional bilinear Schrödinger equation in a bounded domain. In this article, we have given the first available upper bound estimates of the time needed to steer exactly the infinite square potential well from its first eigenstate to the second one.

In [22], we present an embedded automatic strategy for the control of a low consumption vehicle equipped with an “on/off” engine. The proposed strategy has been successfully implemented on the Vir'Volt prototype in official competition (European Shell Eco Marathon).

#### Stabilization of heterogeneous systems

Participants : David Dos Santos Ferreira, Takéo Takahashi, Julie Valein, Jean-Claude Vivalda.

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 [7], we study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure 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.

In [19], we study the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Under appropriate assumption on the coefficients of the feedbacks, we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system, using two different methods: a Lyapunov functional approach and an observability inequality approach.

In [14], we generalize a formula, due to E. Sontag *et al.*, giving explicitly a continuous stabilizing feedback for systems affine in the control; more specifically for a large class of systems which depend quadratically on the control, an explicit formula for a stabilizing feedback law is given.