For a fixed bounded open set $\Omega\subset\mathbb{R}^N$ , a sequence of open sets $\Omega_n\subset\Omega$ and a sequence of sets $\Gamma_n\subset\partial\Omega\cap\partial\Omega_n$ , we study theasymptotic behavior of the solution of a nonlinear ellipticsystem posed on $\Omega_n$ , satisfying Neumann boundary conditionson $\Gamma_n$ and Dirichlet boundary conditions on  $\partial\Omega_n\setminus \Gamma_n$ . We obtain a representationof the limit problem which is stable by homogenization and weprove that this representation depends on $\Omega_n$ and $\Gamma_n$ locally.


In this paper, a $W^{-1,N'}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on $\mathbb R^N$, or on a regular bounded open set of $\mathbb R^N$. The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math. 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc. 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an existence result for the stationary Navier-Stokes equation when the viscosity tensor is only in L1.

The soil water movement modelgoverned by the initial-boundary value problem for a quasilinear1-D parabolic equation with nonlinear coefficients is considered.The generalized statement of the problem is formulated. Thesolvability of the problem is proved in a certain class offunctional spaces. The data assimilation problem for this model isanalysed. The numerical results are presented.

We construct explicitly an homogeneous feedback for a class of single input, two dimensional and homogeneous systems.

We give necessary and sufficient conditions which characterize the Young measures associated to two oscillating sequences of functions, un on $\omega_1\times \omega_2$ and vn on $\omega_2$ satisfying the constraint $v_n(y)=\frac{1}{|\omega_1|} \int_{\omega_1} u_n (x, y) dx$. Our study is motivated by nonlinear effects induced by homogenization. Techniques based on equimeasurability and rearrangements are employed.

We prove existence of minimizing movements for thedislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscositysolutions of the corresponding level-set equation. We also prove theconsistency of this approach, by showing that any minimizing movementcoincides with the smooth evolution as long as the latter exists. Inrelation with this, we finally prove short time existence and uniqueness of a smoothfront evolving according to our law, provided the initial shape issmooth enough.

Dans cet article nous proposonsdifférents algorithmes pour résoudre une nouvelle classe deproblèmes variationels non convexes. Cette classe généraliseplusieurs types d'inégalités variationnelles (Cho etal. (2000), Noor (1992), Zeng (1998), Stampacchia(1964)) du cas convexe au cas non convexe. La sensibilitéde cette classe de problèmes variationnels non convexes a été aussi étudiée.

In this paper we analyze a typical shape optimization problem intwo-dimensional conductivity. We study relaxation for this problemitself. We also analyze the question of the approximation of thisproblem by the two-phase optimal design problems obtained when wefill out the holes that we want to design in the original problemby a very poor conductor, that we make to converge to zero.

This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0+ to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.

We consider a mesoscopic model for phase transitions in a periodic mediumand we construct multibump solutions.The rational perturbative case is dealt with by explicitasymptotics.

In this paper, we investigate the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking as control input the flow of the fluid through a part of the boundary of the rigid body, we obtain a finite-dimensional system similar to Kirchhoff laws in which the control input appears through both linear terms (with time derivative) and bilinear terms. Applying Coron’s return method, we establish some local controllability results for the position and velocities of the underwater vehicle. Examples with six, four, or only three controls inputs are given for a vehicle with an ellipsoidal shape.

Newton's problem of the body of minimal aerodynamic resistance is traditionallystated in the class of convex axially symmetric bodies withfixed length and width. We state and solve the minimal resistanceproblem in the wider class of axially symmetric but generallynonconvex bodies. The infimum in this problem is not attained. Weconstruct a sequence of bodies minimizing the resistance. Thissequence approximates a convex body with smooth front surface, whilethe surface of approximating bodies becomes more and morecomplicated. The shape of the resulting convex body and the value ofminimal resistance are compared with the corresponding results forNewton's problem and for the problem in the intermediate class ofaxisymmetric bodies satisfying the single impact assumption[Comte and Lachand-Robert, J. Anal. Math.83 (2001) 313–335]. In particular, the minimal resistance in our class issmaller than in Newton's problem; the ratio goes to 1/2 as(length)/(width of the body) → 0, and to 1/4 as(length)/(width) → +∞.

In this paper we provide a new sufficiency theorem for regular syntheses. The concept of regular synthesis is discussed in [12], wherea sufficiency theorem for finite time syntheses is proved. There are interesting examples of optimal syntheses that are very regular, but whose trajectories have time domains not necessarilybounded. The regularity assumptions of the main theorem in [12] are verified by every piecewise smooth feedback control generatingextremal trajectories that reach the target in finite time with a finite number of switchings. In the case of this paper the situation iseven more complicate, since we admit both trajectories with finite and infinite time. We use weak differentiability assumptions on the synthesis and weak continuity assumptions on the associated value function.

We give the precise behaviour of some solutions of a nonlinearelliptic B.V.P. in a bounded domain when a parameter approaches aneigenvalue of the principal part. If the nonlinearity has someregularity and the domain is for example convex, we also prove anonlinear version of Courant's Nodal theorem.

This paper is concerned with the internal distributed control problem for the 1D Schrödinger equation, i ut(x,t) = −uxx+α(x) u+m(u) u, that arises in quantum semiconductor models. Here m(u) is a non local Hartree–type nonlinearity stemming from the coupling with the 1D Poisson equation, and α(x) is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the contraction mapping theorem it is shown that for initial and target states belonging to a suitable small neighborhood of the origin, and for distributed controls supported outside of a fixed compact interval, the model equation is controllable. Moreover, it is shown that, for distributed controls with compact support, the exact controllability problem is not possible.

The energy in a square membrane Ω subject to constant viscous dampingon a subset $\omega\subset \Omega$ decays exponentially in timeas soon as ωsatisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min( -\mu(\omega),g(\omega))$ (see Lebeau [Math.Phys. Stud.19 (1996) 73–109]). Here $\mu(\omega)$ denotes the spectral abscissa of thedamped wave equation operator and  $g(\omega)$ is a number called the geometrical quantity of ω and defined as follows.A ray in Ω is the trajectory generated by thefree motion of a mass-point in Ω subject to elastic reflections on theboundary. These reflections obey the law of geometrical optics.The geometrical quantity $g(\omega)$ is then defined as the upper limit (large timeasymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when ωis a finite union of squares.

For smooth or real-analytic single-input, control-affine, non-linear systems, with at least two ouputs, observability for any input ofa given class is generic. This class can be either the class of inputs bounded with their derivatives up to a certain order, or the class ofpolynomial inputs with bounded degree.

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differentialequation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surroundingmanifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold,defined by Rabier and Rheinboldt for determined implicit differential equations, to underdetermined implicit differential equations. With thisgeometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reductionprocedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leadsto, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.

We study the asymptotic behaviour of a sequence of stronglydegenerate parabolic equations $\partial_t (r_h u) - {\rm div}(a_h \cdot Du)$ with $r_h(x,t) \geq0$ , $r_h \in L^{\infty}(\Omega\times (0,T))$ .The main problem is the lack of compactness, by-passed via a regularity result.As particular cases, we obtain G-convergence for elliptic operators $(r_h \equiv 0)$ ,G-convergence for parabolic operators $(r_h \equiv 1)$ , singular perturbationsof an elliptic operator $(a_h \equiv a$ and $r_h \to r$ , possibly $r\equiv 0)$ .

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1)\{½}. It follows that, under this assumption, our starting question has a positive answer.