We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h,k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state. Therefore, it is natural to introduce the controlpoint of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k, in particular the Least Squares method with only one criteria for the pair (h,k) or the Pareto Optimal Control for multicriteria problems. We propose here to use the notion of Hierarchic Control. This notion assumes that we have two controls h, k where h will be the leader while k will be the follower. The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is “adapted” to the constraints. The obtained results are applied to the sentinels theory of Lions [Masson (1992)].

We prove partial regularity with optimal Hölder exponent ofvector-valued minimizers u of the quasiconvex variational integral $\intF( x,u,Du) \,{\rm d}x$ under polynomial growth. We employ the indirectmethod of the bilinear form.

It is shown that self-locomotion is possible for a body in Euclidian space,provided its dynamics corresponds to a non-quadratic Hamiltonian,and that the body contains at least 3 particles. The efficiencyof the driver of such a system is defined. The existence of anoptimal (most efficient) driver is proved.


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 $r\equiv 0)$ .


We prove a $C^{k,\alpha}$ partial regularity result for local minimizers of variationalintegrals of the type $I(u)=\int_\Omega f(D^{k}u(x)){\rm d}x$ , assumingthat the integrand f satisfies (p,q) growth conditions.


The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\textrm{div}\,a(\nabla u)+F[u](x)=0,$ over the functions $u\in W^{1,1}(\Omega)$ that assume given boundary values ϕ on ∂Ω. The vector field $a:{\mathbb R}^n\to {\mathbb R}^n$ satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions when ϕ satisfies the classical bounded slope condition. In a variational context, Clarke [Ann. Sc. Norm. Super. Pisa Cl. Sci.4 (2005) 511–530] has introduced a new type of hypothesis on the boundary condition ϕ: the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if ϕ is the restriction to ∂Ω of a convex function. We show that if a and F satisfy hypotheses similar to those of Hartman and Stampacchia, the lower bounded slope condition implies the existence of solutions in the space of locally Lipschitz continuous functions on Ω.


We consider, in an open subset Ω of ${\mathbb R}^N$ , energies depending on the perimeter of a subset $E\subset\Omega$ (or some equivalent surface integral) and on a function u which isdefined only on $\Omega\setminus E$ . We compute the lower semicontinuous envelopeof such energies. This relaxation has to take intoaccount the fact that in the limit, the “holes” E maycollapse into a discontinuity of u, whose surface will be countedtwice in the relaxed energy. We discuss some situations where suchenergies appear, and give, as an application, a new proofof convergence for an extensionof Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.


We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbedconvection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on themacroscopic scale and on the periodic microscopic scale. Denoting by ε the period, the potential or zero-orderterm is scaled as $\varepsilon^{-2}$ and the drift or first-order term is scaled as $\varepsilon^{-1}$ . Under a structuralhypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain withnon-degenerate quadratic behavior, we prove an exponential localization at this minimum point. The homogenizedproblem features a diffusion equation with quadratic potential in the whole space.

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during themodeling of diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local optimality in an Ls -neighborhood of a reference function whereby the underlying analysis allows to use weaker norms than $L^\infty$ .

An open-loop system of a multidimensional wave equationwith variable coefficients, partial boundary Dirichlet control andcollocated observation is considered. It is shown that the system iswell-posed in the sense of D. Salamon and regular in the sense of G.Weiss. The Riemannian geometry method is used in the proof ofregularity and the feedthrough operator is explicitly computed.

We consider the Laplace operator in a thin tube of ${\mathbb R}^3$ with a Dirichlet condition on its boundary. We study asymptotically the spectrum ofsuch an operator as the thickness of the tube's cross section goes to zero. In particular weanalyse how the energy levels depend simultaneously on the curvature of the tube's central axisand on the rotation of the cross section with respect to the Frenet frame. The main argument is aΓ-convergence theorem for a suitable sequence of quadratic energies.

We study the stability of a sequence of integralfunctionals on divergence-free matrix valued fields following the directmethods of Γ-convergence. We prove that the Γ-limitis an integral functional on divergence-free matrix valued fields.Moreover, we show that the Γ-limit is also stable undervolume constraint and various type of boundary conditions.