A Grobman–Hartman Theorem for Control Systems

We consider the problem of locally linearizing a control system via topological transformations. According to [2,3], there is no naive generalization of the classical Grobman–Hartman theorem for ODEs to control systems: a generic control system, when viewed as a set of under-determined differential equations parametrized by the control, cannot be linearized using pointwise transformations on the state and the control values. However, if we allow the transformations to depend on the control at a functional level (open loop transformations), we are able to prove a version of the Grobman–Hartman theorem for control systems.     

Construction d'un ensemble d-dominant sur un graphe en utilisant un critère donnéConstruction of a d-dominating set on a graph by using a given criterion

d-dominating sets in graphs are very important in system and network engineering. Their constructions is thus an important research topic. An heuristic which forms such sets by using a given criterion is proposed. We simplify the heuristic presented elsewhere. We extend it and prove the correctness of the extended heuristic. This heuristic has the advantage of being distributed and scalable. To cite this article: A. Delye de Clauzade de Mazieux et al., C. R. Mecanique 334 (2006).     

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

The notion of Loschmidt echo (also called “quantum fidelity”) has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of “quantum chaos”, quantum computation and decoherence. In this paper, we study this quantity semiclassically (as ), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called “Ehrenfest time” , we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the “return probability” (again on reference states being the coherent states) by the same method, as a function of t and . Submitted: April 27, 2006; Accepted: May 11, 2006     

Connections between semidefinite relaxations of the max-cut and stable set problems

We describe links between a recently introduced semidefinite relaxation for the max-cut problem and the well known semidefinite relaxation for the stable set problem underlying the Lovász’s theta function. It turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra. We also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations. This work was done while the author visited CWI, Amsterdam, The Netherlands. Svata Poljak untimely deceased in April 1995. We shall both miss Svata very much. Svata was an excellent colleague, from whom we learned a lot of mathematics and with whom working was always a very enjoyable experience; above all, Svata was a very nice person and a close friend of us. The research was partly done while the author visited CWI, Amsterdam, in October 1994 with a grant fom the Stieltjes Institute, whose support is gratefully acknowledged. Partially supported also by GACR 0519. Research support by Christian Doppler Laboratorium für Diskrete Optimierung.     

A generalized flat extension theorem for moment matrices

In this note we prove a generalization of the flat extension theorem of Curto and Fialkow (Memoirs of the American Mathematical Society, vol. 119. American Mathematical Society, Providence, 1996) for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.      

Review of inventory systems with deterioration since 2001

This paper presents an up-to-date review of the advances made in the field of inventory control of perishable items (deteriorating inventory). The last extensive review on this topic dates back to 2001 (Goyal S.K. and Giri B.C., Recent trends in modeling of deteriorating inventory, European Journal of Operational Research, 134, 1–16). Since then, over two hundred articles on this subject have been published in the major journals on inventory control, indicating the need for a new review. We use the classification of Goyal and Giri based on shelf life characteristics and demand characteristics. Contributions are highlighted by discussing main system characteristics, including price discounts, backordering or lost sales, single or multiple items, one or two warehouses, single or multi-echelon, average cost or discounted cash flow, and payment delay.     

Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme

We study a three-dimensional model for the skin effect in electromagnetism. We first give a multiscale asymptotic expansion for the solution of the harmonic Maxwell equations set on a domain made of two materials, dielectric and highly conducting, with a regular interface between them. To measure the skin effect, we introduce a suitable skin depth function defined on the interface and generalizing the classical scalar quantity. We then prove an asymptotic expansion at high conductivity for this function, which exhibits the influence of the geometry of the interface on the skin depth.     

Asymptotics for the Low-Lying Eigenstates of the Schrödinger Operator with Magnetic Field near Corners

The Neumann realization for the Schr?dinger operator with magnetic field is considered in a bounded two-dimensional domain with corners. This operator is associated with a small semi-classical parameter h or, equivalently, with a large magnetic field.We investigate the behavior of its eigenpairs as h tends to zero, like in a semi-classical limit. We prove, in the situation where the domain is a polygon and the magnetic field is constant, that the lowest eigenvalues are exponentially close to those of model problems associated with the corners. We approximate the corresponding eigenvectors by linear combinations of functions concentrated in corners at the scale If the domain has curved sides and the magnetic field is smoothly varying, we exhibit a full asymptotics for eigenpairs in powers of Communicated by Christian Gérard Submitted: October 13, 2005 Accepted: December 19, 2005