We consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

A variant of the Total OverlappingSchwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1Math.336(2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains.That same method turns to be an efficient toolto make numerical zoomsin regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture andthe reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper isto use this modified Schwarz procedure as a preconditioner to Krylovsubspaces methods so to accelerate the calculations. A detailed study concludes toa super-linear convergence ofGMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numericalexamples are also provided and commented that demonstrate the reliability of theTOS-preconditioner.

While alternans in a single cardiac cell appears through a simpleperiod-doubling bifurcation, in extended tissue the exact natureof the bifurcation is unclear. In particular, the phase ofalternans can exhibit wave-like spatial dependence, eitherstationary or travelling, which is known as discordantalternans. We study these phenomena in simple cardiac modelsthrough a modulation equation proposed by Echebarria-Karma. Asshown in our previous paper, the zero solution of their equationmay lose stability, as the pacing rate is increased, througheither a Hopf or steady-state bifurcation. Which bifurcationoccurs first depends on parameters in the equation, and for onecritical case both modes bifurcate together at a degenerate(codimension 2) bifurcation. For parameters close to thedegenerate case, we investigate the competition between modes,both numerically and analytically. We find that at sufficientlyrapid pacing (but assuming a 1:1 response is maintained), steadypatterns always emerge as the only stable solution. However, inthe parameter range where Hopf bifurcation occurs first, theevolution from periodic solution (just after the bifurcation) tothe eventual standing wave solution occurs through an interestingseries of secondary bifurcations.

We analyze the accuracy and well-posedness of generalized impedanceboundary value problems in the framework of scattering problemsfrom unbounded highly absorbing media. We restrict ourselves in this first workto the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficultiesin the analysis for the generalized impedance boundary conditions, sinceclassical compactness arguments are no longer possible. Our new analysisis based on the use of Rellich-type estimates and boundedness of L2solution operators. We also discuss some numerical experimentsconcerning these boundary conditions.

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditionsto obtain fast convergence of domain decomposition methods.After explaining the theoretical results, we explicitly compute the coefficients in the transmission boundary conditions. The numerical results presented in this paper confirm the optimality properties.

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutionsof polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ $\mathbb{R}^d$ , d = 2 or 3, for the velocity andthe pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function $\beta^L(\cdot) :=\min(\cdot,L)$ in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker–Planck–Navier–Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions.We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms,and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

We consider the P n model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation.We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale.The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P 1 model without absorption term. It requires the computation of the solution of the steady state P n equations. This is made by one Monte-Carlo simulation performed outside the time loop. Using formal expansions with respect to a small parameter representing the inverse of the number of mean free path in each cell, the resulting scheme is proved to have the diffusion limit. In order to avoid the CFL constraint on the time step, we give an implicit version of the scheme which preserves the positivity of the zeroth moment.

We prove that any Kantorovich potential for the cost functionc = d2/2 on a Riemannian manifold (M, g) is locally semiconvexin the “region of interest”, without any compactness assumptionon M, nor any assumption on its curvature. Such a region ofinterest is of full μ-measure as soon as the starting measureμ does not charge n – 1-dimensional rectifiable sets.


In this paper, the stability of a Timoshenko beam with time delaysin the boundary input is studied. The system is fixed at the leftend, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system iswell-posed. By the complete spectral analysis, we show that there isa sequence of eigenvectors and generalized eigenvectors of thesystem operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.


The Monge-Kantorovich problem is revisited by means of a variantof the saddle-point method without appealing to c-conjugates. Anew abstract characterization of the optimal plans is obtained inthe case where the cost function takes infinite values. It leadsus to new explicit sufficient and necessary optimality conditions.As by-products, we obtain a new proof of the well-knownKantorovich dual equality and an improvement of the convergence ofthe minimizing sequences.

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.

The paper represents the first part of a series ofpapers on realization theory of switched systems. Part I presents realization theory of linear switched systems,Part II presents realization theory of bilinear switched systems.More precisely, in Part I necessary and sufficient conditionsare formulated for a family of input-output maps to berealizable by a linear switched system and a characterizationof minimal realizations is presented. The paper treats two types of switched systems.The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences isadmissible, but within this restricted set the switching times are arbitrary.The paper uses the theory of formal power series to derivethe results on realization theory.

We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original problem, and admits a global minimizer. As an application, we provide existence theorems for the image registration problem with convex and polyconvex regularization terms.

The topological asymptotic analysis provides the sensitivity of a givenshape functional with respect to an infinitesimal domain perturbation, likethe insertion of holes, inclusions, cracks. In this work we present thecalculation of the topological derivative for a class of shape functionalsassociated to the Kirchhoff plate bending problem, when a circular inclusionis introduced at an arbitrary point of the domain. According to theliterature, the topological derivative has been fully developed for a widerange of second-order differential operators. Since we are dealing here witha forth-order operator, we perform a complete mathematicalanalysis of the problem.

This paper is the second part of a series of papers dealing withrealization theory of switched systems.The current Part II addresses realization theory of bilinear switchedsystems. In Part I [Petreczky, ESAIM: COCV, DOI: 10.1051/cocv/2010014] we presented realizationtheory of linear switched systems.More precisely, in Part II we present necessary and sufficient conditionsfor a family of input-output maps to berealizable by a bilinear switched system, together with a characterizationof minimal realizations.Similarly to Part I, the paper deals with two types of switched systems.The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences isadmissible, but within this restricted set the switching times are arbitrary.The paper uses the theory of formal power series to derivethe results on realization theory.

We propose a general reduced-order filtering strategy adapted to Unscented Kalman Filtering for any choice of sampling points distribution. This provides tractable filtering algorithms which can be used with large-dimensional systems when the uncertainty space is of reduced size, and these algorithms only invoke the original dynamical and observation operators, namely, they do not require tangent operator computations, which of course is of considerable benefit when nonlinear operators are considered. The algorithms are derived in discrete time as in the classical UKF formalism – well-adapted to time discretized dynamical equations – and then extended into consistent continuous-time versions. This reduced-order filtering approach can be used in particular for the estimation of parameters in large dynamical systems arising from the discretization of partial differential equations, when state estimation can be handled by an adequate Luenberger observer inspired from feedback control. In this case, we give an analysis of the joint state-parameter estimation procedure based on linearized error, and we illustrate the effectiveness of the approach using a test problem inspired from cardiac biomechanics.

This paper focuses on the analytical properties of the solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with respect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows.

The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the time-minimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.