We present equivalent conditions and asymptotic models for the diffraction problem ofelastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium.Due to the thinness of the layer with respect to the wavelength, this problem is wellsuited for the notion of equivalent conditions and the effect of the fluid medium on thesolid is as a first approximation local. We derive and validate equivalent conditions upto the fourth order for the elastic displacement. These conditions approximate theacoustic waves which propagate in the fluid region. This approach leads to solve onlyelastic equations. The construction of equivalent conditions is based on a multiscaleexpansion in power series of the thickness of the layer for the solution of thetransmission problem.

We show that we can reconstruct two coefficients of a wave equation by a single boundary measurement of the solution. The identification and reconstruction are based on Krein’s inverse spectral theory for the first coefficient and on the Gelfand−Levitan theory for the second. To do so we use spectral estimation to extract the first spectrum and then interpolation to map the second one. The control of the solution is also studied.

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to Acerbi and Fusco combined with a Meyers Lp-estimate adapted to the functional ellipticity condition. The present framework includes in particular the elasticity case and the reinforcement by stiff thin fibers.

This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0+ to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of the calculus of variations are described. Necessary optimality conditions in the main problem of the calculus of variations and in the problem of Bolza in the nonsmooth case are derived. Examples comparing presented results with other approaches to nonsmooth problems of the calculus of variations are given.

We prove the continuity and the Hölder equivalence w.r.t. an Euclidean distance of the value function associated with the L1 cost of the control-affine system q̇ = f0(q) + ∑j=1m uj fj(q), satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball–Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the L1 cost of time-dependent systems of the form q̇ = ∑j=1m uj fjt(q).

We introduce the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external policy maker, and we propagate its effect for the number N of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This will be realized by considering cost functionals including L1-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of Γ-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.

The steady response of a fluid with two layers of different density in a porous medium is considered during extraction through a point sink. Supercritical withdrawal in which both layers are being withdrawn is investigated using a spectral method. We show that for each withdrawal rate, there is a single entry angle of the interface into the point sink. As the flow rate decreases the angle of entry steepens until it becomes almost vertical, at which point the method fails. This limit is shown to correspond to the upper bound on sub-critical (single-layer) flow.

A phase field approach for structural topology optimization which allows for topologychanges and multiple materials is analyzed. First order optimality conditions arerigorously derived and it is shown via formally matched asymptoticexpansions that these conditions converge to classical first order conditions obtained inthe context of shape calculus. We also discuss how to deal with triple junctions wheree.g. two materials and the void meet. Finally, we present severalnumerical results for mean compliance problems and a cost involving the least square errorto a target displacement.

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem.

We study the first eigenpair of a Dirichlet spectral problem for singularly perturbedconvection-diffusion operators with oscillating locally periodic coefficients. It followsfrom the results of [A. Piatnitski and V. Rybalko, On the first eigenpair of singularlyperturbed operators with oscillating coefficients. Preprintwww.arxiv.org, arXiv:1206.3754] that thefirst eigenvalue remains bounded only if the integral curves of the so-called effectivedrift have a nonempty ω-limit set. Here we consider the case when theintegral curves can have both hyperbolic fixed points and hyperbolic limit cycles. One ofthe main goals of this work is to determine a fixed point or a limit cycle responsible forthe first eigenpair asymptotics. Here we focus on the case of limit cycles that was leftopen in [A. Piatnitski and V. Rybalko, Preprint.

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependent damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings’ inherent difficulties.

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases impossible to do within a reasonable time. We propose here a very simple methodology which may accelerate population parametrization of complex models, including partial differential equations models. We illustrate our method on the classical KPP equation.

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤−1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189–222]. We also get the corrector results.

In this paper, we give general recommendations for successful application of the Douglas–Rachford reflection method to convex and nonconvex real matrix completion problems. These guidelines are demonstrated by various illustrative examples.

This paper focuses on a one-dimensional wave equation being subjected to a unilateralboundary condition. Under appropriate regularity assumptions on the initial data, a newproof of existence and uniqueness results is proposed. The mass redistribution method,which is based on a redistribution of the body mass such that there is no inertia at thecontact node, is introduced and its convergence is proved. Finally, some numericalexperiments are reported.

We present in this paper the formal passage from a kinetic model to the incompressible Navier−Stokes equations for a mixture of monoatomic gases with different masses. The starting point of this derivation is the collection of coupled Boltzmann equations for the mixture of gases. The diffusion coefficients for the concentrations of the species, as well as the ones appearing in the equations for velocity and temperature, are explicitly computed under the Maxwell molecule assumption in terms of the cross sections appearing at the kinetic level.