Let ${\cal L}$ (x,u,∇ u) be a Lagrangian periodic of period 1 in x 1,...,x n ,u. We shall study the non self intersectingfunctions u: R n ${\to}$ R minimizing ${\cal L}$ ; non self intersecting means that, if u(x 0 + k) + j = u(x 0) for some x 0∈R n and (k , j) ∈Z n × Z, then u(x) = u(x + k) + j $\;\forall$ x. Moser has shown that each of these functions is at finite distance from a plane u = ρ $\cdot$ x and thus has an average slope ρ; moreover, Senn has proven that it is possible to define the average action of u, which is usually called $\beta(\rho)$ since it only depends on the slope of u.Aubry and Senn have noticed a connection between $\beta(\rho)$ and the theory of crystals in ${\bf R}^{n+1}$ , interpreting $\beta(\rho)$ as the energy per area of a crystal face normal to $(-\rho,1)$ . The polar of β is usually called -α; Senn has shown that α is C 1 and that the dimension of the flat of α which contains c depends only on the “rational space” of $\alpha^\prime$ (c). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of α: they are C 1 and their dimension depends only on the rational space of their normals.


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.


Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K $\subset \mathbb{R}^{nm}$ instead of the whole space $\mathbb{R}^{nm}$ as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope $f^{(qc)} (v) = {\rm sup} \{ \,g(v)\, \vert \,g : \mathbb{R}^{nm} \rightarrow \mathbb{R} \cup \{ + \infty \}$ quasiconvex and lower semicontinuous, $g(v) \leq f(v) \,\,\,\,\forall v \in \mathbb{R}^{nm}\,\}.$ Our main result is a representation theorem for $f^{({\it qc})}$ which generalizes Dacorogna's well-known theorem on the representation of the quasiconvex envelope of a finite function. The paper will be completed by the calculation of $f^{({\it qc})}$ in two examples.


We consider an optimal control problem for a system of the form $\dot{x}$ = f(x,u), with a running cost L. We prove an interiorsphere property for the level sets of the corresponding valuefunction V. From such a property we obtain a semiconcavityresult for V, as well as perimeter estimates for the attainablesets of a symmetric control system.

This paper is concerned with mathematical modelling in the management of a wastewater treatment system. The problem is formulated as looking for a Nash equilibrium of a multiobjective pointwise control problem of aparabolic equation. Existence of solution is proved and a first order optimality system is obtained. Moreover, a numerical method to solve this system is detailed and numerical results are shown in a realistic situation posed in the estuary of Vigo (Spain).


An alternative approach for the analysis and the numericalapproximation of ODEs, using a variational framework, ispresented. It is based on the natural and elementary idea of minimizing the residual of the differential equation measured in a usual Lp norm.Typical existence results for Cauchy problems can thus berecovered, and finer sets of assumptions for existence are made explicit. We treat, in particular, the cases of an explicit ODE and a differential inclusion. This approach also allows for a whole strategy to approximate numerically the solution. It is briefly indicated here as it will be pursued systematically and in a much more broad fashion in a subsequent paper.

The purpose of this paper is to derive and study a new asymptoticmodel for the equilibrium state of a thin anisotropicpiezoelectric plate in frictional contact with a rigid obstacle.In the asymptotic process, the thickness of the piezoelectricplate is driven to zero and the convergence of the unknowns isstudied. This leads to two-dimensional Kirchhoff-Love plateequations, in which mechanical displacement and electric potentialare partly decoupled. Based on this model numerical examples arepresented that illustrate the mutual interaction between themechanical displacement and the electric potential. We observethat, compared to purely elastic materials, piezoelectric bodiesyield a significantly different contact behavior.

We study Hamiltonian systems which generate extremal flows of regularvariational problems on smooth manifolds and demonstrate thatnegativity of the generalized curvature of such a system impliesthe existence of a global smooth optimal synthesis for the infinitehorizon problem.We also show that in the Euclidean case negativity of the generalized curvature is a consequence ofthe convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems.

A general framework for computing robust controllable sets ofconstrained nonlinear uncertain discrete-time systems as well ascontrolling such complex systems based on the computed robustcontrollable sets is introduced in this paper. The addressedone-step control approach turns out to be a robust model predictivecontrol scheme with feasible unit control horizon and contractiveconstraint. The solver of 1-dimensional quantified set inversion inmodal interval analysis is extended to 2-dimensional cases forcomputing robust controllable sets off-line with a clear semanticinterpretation, where both universal and existential quantifiers areconcerned simultaneously. An interval-based solver of constrainedminimax optimization is also proposed to compute one-step controlinputs online in a reliable way, which guarantee to drive the systemstate contractively along the computed robust controllable sets to aselected terminal robust control invariant set.

A direct construction of a stabilizing hybrid feedback that is robust togeneral measurement error is given for a general nonlinear control system that is asymptotically controllable to a compact set.

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.