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Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set

Luciano Carbone, Doina Cioranescu, Riccardo De Arcangelis, Antonio Gaudiello (2004)

ESAIM: Control, Optimisation and Calculus of Variations

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set

Luciano Carbone, Doina Cioranescu, Riccardo De Arcangelis, Antonio Gaudiello (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses...

Homogenization of variational problems in manifold valued Sobolev spaces

Jean-François Babadjian, Vincent Millot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, a Γ-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation...

Homogenization with uncertain input parameters

Luděk Nechvátal (2010)

Mathematica Bohemica

We homogenize a class of nonlinear differential equations set in highly heterogeneous media. Contrary to the usual approach, the coefficients in the equation characterizing the material properties are supposed to be uncertain functions from a given set of admissible data. The problem with uncertainties is treated by means of the worst scenario method, when we look for a solution which is critical in some sense.

How humans fly

Alain Ajami, Jean-Paul Gauthier, Thibault Maillot, Ulysse Serres (2013)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular classes...

How to state necessary optimality conditions for control problems with deviating arguments?

Lassana Samassi, Rabah Tahraoui (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: inf ( u , v ) ∈ 𝒰 a d ∫ 0 1 f t , u ( θ v ( t ) ) , u ' ( t ) , v ( t ) d t , (1) where 𝒰 a d is a set of admissible controls and θ v is the solution of the following equation: { d θ ( t ) d t = g ( t , θ ( t ) , v ( t ) ) , t ∈ [ 0 , 1 ] ; θ ( 0 ) = θ 0 , θ ( t ) ∈ [ 0 , 1 ] ∀ t . (2). The results are nonlocal and new.

Identification of critical curves. II. Discretization and numerical realization

Jaroslav Haslinger, Václav Horák, Pekka Neittaanmäki, Kimmo Salmenjoki (1991)

Applications of Mathematics

We consider the finite element approximation of the identification problem, where one wishes to identify a curve along which a given solution of the boundary value problem possesses some specific property. We prove the convergence of FE-approximation and give some results of numerical tests.

Identification problem for nonlinear beam -- extension for different types of boundary conditions

Radová, Jana, Machalová, Jitka (2023)

Programs and Algorithms of Numerical Mathematics

Identification problem is a framework of mathematical problems dealing with the search for optimal values of the unknown coefficients of the considered model. Using experimentally measured data, the aim of this work is to determine the coefficients of the given differential equation. This paper deals with the extension of the continuous dependence results for the Gao beam identification problem with different types of boundary conditions by using appropriate analytical inequalities with a special...

Image segmentation with a finite element method

Blaise Bourdin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The Mumford-Shah functional for image segmentation is an original approach of the image segmentation problem, based on a minimal energy criterion. Its minimization can be seen as a free discontinuity problem and is based on Γ-convergence and bounded variation functions theories. Some new regularization results, make possible to imagine a finite element resolution method. In a first time, the Mumford-Shah functional is introduced and some existing results are quoted. Then, a discrete formulation...

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