In this paper, we prove a controllabilityresult for a fluid-structure interaction problem. In dimension two,a rigid structure moves into an incompressible fluid governed byNavier-Stokes equations. The control acts on a fixed subset of thefluid domain. We prove that, for small initial data, this system isnull controllable, that is, for a given T > 0, the system can bedriven at rest and the structure to its reference configuration attime T. To show this result, we first consider a linearizedsystem. Thanks to an observability inequality obtained from aCarleman inequality, we prove an optimal controllability result witha regular control. Next, with the help of Kakutani's fixed pointtheorem and a regularity result, we pass to the nonlinear problem.

The goal of this paper is to study the so-called worst-case or robustoptimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst,compliance when the loads are subject to some unknown perturbations.We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm tocompute it. Then, in the framework of Hadamard method, wecompute the directional shape derivative of this criterion which isused in a numerical algorithm, based on the level set method,to find optimal shapes that minimize the worst-case compliance.Since this criterion is usually merely directionally differentiable,we introduce a semidefinite programming approach toselect the best descent direction at each step of agradient method. Numerical examples are given in 2-d and 3-d.

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\{\nabla u_k\}$ , bounded in $L^p(\Omega;{\mathbb R}^{m\times n})$ if p > 1 and $\Omega\subset{\mathbb R}^n$ is a bounded domain with the extension property in $W^{1,p}$ .Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Caseswhere no boundary conditions nor regularity of Ω arerequired and links with lower semicontinuity results by Meyers andby Acerbi and Fusco are also discussed.

We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given $\phi_{0}$ close enough to an eigenstate corresponding to the length l = 1 and $\phi_{f}$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb{R}^{*}_{+}$ with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_{0}$ to $\phi_{f}$ in time T.In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way.Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order.

Let $F^{\rm p} \in {\rm GL}(3)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form ${\rm Curl}[{F^{\rm p}}]\cdot (F^{\rm p})^T$ applied to rotations controls the gradient in the sense that pointwise $ \forall R \in C^1(\mathbb{R}^3, {\rm SO}(3)): \Arrowvert {\rm Curl}[R] \cdot R^T \Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}R\Arrowvert_{\mathbb{R}^{27}}^2$ .This result complements rigidity results[Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying that $ \forall A \in C^1(\mathbb{R}^3, \mathfrak{so}(3)): \Arrowvert {\rm Curl}[A]\Arrowvert_{\mathbb{M}^{3\times3}}^2 \ge \frac{1}{2} \Arrowvert{\rm D}A\Arrowvert_{\mathbb{R}^{27}}^2 = \Arrowvert\nabla{\rm axl}[A]\Arrowvert_{\mathbb{R}^9}^2$ .

The gradient based topological optimization tools introduced during thelast ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remainunchanged, at least during the topological phase of the optimizationalgorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not forimproving the cost function. Then, two numerical examples are presented:the first one compares topological optimization with standard shape optimization, and the second one, issued from a lake oxygenation problem, illustrates the use of the new asymptotic expansion.


For a class of 2-D elastic energies we show that a radial equilibrium solutionis the unique global minimizer in a subclass of all admissible maps. Theboundary constraint is a double cover of S 1; the minimizer u is C 1and is such that $\det\nabla u$ vanishes at one point.