The aim of this article is to present algorithms to compute the firstconjugate time along a smooth extremal curve, where the trajectoryceases to be optimal. It is based on recent theoretical developmentsof geometric optimal control, and the article contains a reviewof second order optimality conditions.The computations are related to a testof positivity of the intrinsic second order derivative or a test ofsingularity of the extremal flow. We derive an algorithm called COTCOT(Conditions of Order Two and COnjugate Times), available on the web,and apply it to the minimal time problem of orbit transfer, and to theattitude control problem of a rigid spacecraft.This algorithm involves both normal and abnormal cases.

We consider the linearized elasticity system in a multidomain of ${\bf R}^3$ . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius $r^{\varepsilon}$ . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and $r^{\varepsilon}$ tend to zero simultaneously, with $r^{\varepsilon}\gg\varepsilon^2$ , we identify the limit problem. This limit problem involves six junction conditions.

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)].

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms  and characterize them in the rotationally invariant jointly rank-r convex case.

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection ofeither (a) a hysteretic input nonlinearity, an L 2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L 2-stable, time-invariantlinear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certainconstant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system.

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.

This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and two controls. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about “(x,u)-flatness” of these systems, with much more elementary techniques.

This paper gives a rigorous derivationof a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filamentsin a rotationally forced Bose-Einstein condensate. Thisfunctional is derived as a Γ-limitof scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence of large numbers of nontrivial local minimizers and we prove that, given any suchlocal minimizer, the Gross-Pitaevsky functionalhas a local minimizer that is nearby (in a suitable sense) whenever a scalingparameter is sufficiently small.

In this paper we prove a comparison result between semicontinuousviscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_t+H(x,Du) = 0$ in ${\rm I}\!{\rmR}^n\times(0,T)$ where the Hamiltonian H may be noncoercive inthe gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

We study a variational problem which was introduced by Hannon,Marcus and Mizel [ESAIM: COCV9 (2003) 145–149] todescribe step-terraces on surfaces of so-called “unorthodox” crystals.We show that there is no nondegenerate intervals on which the absolutevalue of a minimizer is $\pi/2$ identically.

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.


This work is devoted to theanalysis of a viscous finite-difference space semi-discretizationof a locally damped wave equation in a regular 2-D domain. Thedamping term is supported in a suitable subset of the domain, sothat the energy of solutions of the damped continuous waveequation decays exponentially to zero as time goes to infinity.Using discrete multiplier techniques, we prove that adding asuitable vanishing numerical viscosity term leads to a uniform(with respect to the mesh size) exponential decay of the energyfor the solutions of the numerical scheme. The numerical viscosityterm damps out the high frequency numerical spurious oscillationswhile the convergence of the scheme towards the original dampedwave equation is kept, which guarantees that the low frequenciesare damped correctly. Numerical experiments are presented andconfirm these theoretical results. These results extend those byTcheugoué-Tébou and Zuazua [Numer. Math.95, 563–598 (2003)] where the 1-D casewas addressed as well the square domain in 2-D. The methods andresults in this paper extend to smooth domains in any spacedimension.

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 $r\equiv 0)$ .

In this paper, weconsider probability measures μ and ν on a d-dimensionalsphere in ${\bf R}^{d+1}, d \geq 1,$ and cost functions of the form $c({\bf x},{\bf y})=l(\frac{|{\bf x}-{\bf y}|^2}{2})$ that generalize those arising in geometric optics where $l(t)=-\log t.$ We prove that if μ and ν vanish on $(d-1)$ -rectifiable sets,if |l'(t)|>0, $\lim_{t\rightarrow 0^+}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then thereexists a unique optimal map T o that transports μ onto $\nu,$ whereoptimality is measured against c. Furthermore, $\inf_{{\bf x}}|T_o{\bf x}-{\bf x}|>0.$ Our approach is based on direct variational arguments.In the special case when $l(t)=-\log t,$ existence of optimal maps on thesphere was obtained earlier in [Glimm and Oliker, J. Math. Sci.117 (2003) 4096-4108]and [Wang, Calculus of Variations and PDE's20 (2004) 329-341] under more restrictive assumptions. In these studies, it was assumed that either μ and ν are absolutelycontinuous with respect to the d-dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interestin this work is that it is in contrast with the work in[Gangbo and McCann, Quart. Appl. Math.58 (2000) 705-737] where it is proved that when l(t)=t thenexistence of an optimal map fails when μ and ν are supported by Jordan surfaces.

In this paper we find the optimal regularity for viscositysolutions of the pseudo infinity Laplacian. We prove that thesolutions are locally Lipschitz and show an example that provesthat this result is optimal. We also show existence and uniquenessfor the Dirichlet problem.

We consider the approximation of a class ofexponentially stable infinite dimensional linear systems modellingthe damped vibrations of one dimensional vibrating systems or ofsquare plates. It is by now well known that the approximatingsystems obtained by usual finite element or finite difference arenot, in general, uniformly stable with respect to the discretizationparameter. Our main result shows that, by adding a suitablenumerical viscosity term in the numerical scheme, our approximationsare uniformly exponentially stable. This result is then applied toobtain strongly convergent approximations of the solutions of thealgebraic Riccati equations associated to an LQR optimal controlproblem. We next give an application to a non-homogeneous stringequation. Finally we apply similar techniques for approximating theequations of a damped square plate.

A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zerowhose support is spread out overthe real line.


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.


We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.

We study the homogenization process of ferromagnetic multilayers in the presence of surface energies:super-exchange, also called interlayer exchange coupling,and surface anisotropy. The two main difficulties are the non-linearity of the Landau-Lifshitz equation and the absence of a good sequenceof extension operators for the multilayer geometry.First, we consider the case when surface anisotropy is the dominant term, then the case when the magnitude of the super-exchange interaction is inversely proportional to the interlayer distance. We establishthe homogenized equation in these two situations.