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.

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 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.

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 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.

We propose a necessary and sufficient condition about the existence of variations, i.e.,of non trivial solutions $\eta\in W^{1,\infty}_0(\Omega)$ to the differential inclusion $\nabla\eta(x)\in-\nabla u(x)+{\bf D}$ .

We show that local minimizers of functionals of the form $\int_{\Omega} \left[f(Du(x)) + g(x\,,u(x))\right]\,{\rm d}x$ ,  $u \in u_0 + W_0^{1,p}(\Omega)$ ,are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

In this article, we build a mathematical model to understand theformation of a tree leaf. Our model is based on the idea that a leaftends to maximize internal efficiency by developing an efficienttransport system for transporting water and nutrients. The meaningof “the efficient transport system” may vary as the type of thetree leave varies. In this article, we will demonstrate that treeleaves have different shapes and venation patterns mainly becausethey have adopted different efficient transport systems.The efficient transport system of a tree leaf built here is amodified version of the optimal transport path, which was introducedby the author in [Comm. Cont. Math.5 (2003) 251–279; Calc. Var. Partial Differ. Equ.20 (2004) 283–299; Boundary regularity of optimal transport paths,Preprint] to study thephenomenon of ramifying structures in mass transportation. In thepresent paper, the cost functional on transport systems iscontrolled by two meaningful parameters. The first parameterdescribes the economy of scale which comes with transporting large quantities together, while thesecond parameter discourages the direction of outgoing veins at eachnode from differing much from the direction of the incoming vein.Under the same initial condition, efficient transport systemsmodeled by different parameters will provide tree leaves withdifferent shapes and different venation patterns.Based on this model, we also provide some computer visualization oftree leaves, which resemble many known leaves including the mapleand mulberry leaf. It demonstrates that optimal transportation playsa key role in the formation of tree leaves.

This paper considers the inversion problem related to themanipulation of quantumsystems using laser-matter interactions. The focusis on the identification of the field free Hamiltonian and/orthe dipole moment of aquantum system. The evolution of the system is given by the Schrödingerequation. The available data are observations as a function of timecorresponding to dynamics generated by electric fields. Thewell-posedness of the problem is proved, mainly focusing on the uniqueness ofthe solution. A numerical approach is also introduced with anillustration of its efficiency on a test problem.

We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

This errata corrects one error in the 2004 version of this paper [Mennucci, ESAIM: COCV10 (2004) 426–451].