Daniel Goeleven

University of La Réunion · Laboratory of Physics and Mathematical Engineering for Energy and the Environment (PIMENT)

A semi-discretized scheme is applied to a two-phase free boundary problem issued from the thermal behavior in phase-change materials. The numerical method is based on backward Euler scheme. A priori estimates for the solution of the semi-discretized problem are stated and convergence to the solution of the continuous problem is studied. The existen...

It is well known that modeling friction forces is a complex problem and constitutes an important topic in both mechanical engineering and applied mathematics. In this paper, we show how the approach of Moreau and Panagiotopoulos can be used to develop a suitable methodology for the formulation and the mathematical analysis of various friction model...

Many physical phenomena can be modeled as a feedback connection of a linear dynamical systems combined with a nonlinear function which satisfies a sector condition. The concept of absolute stability, proposed by Lurie and Postnikov (Appl Math Mech 8(3), 1944) in the early 1940s, constitutes an important tool in the theory of control systems. Lurie...

Spitting spiders (Scytodes sp.) spit a mixture of silk and glue at their prey during attack. In this note, we show that a nonsmooth oscillator can be used as a biomechanical model to describe the zig-zag patterns produced by the spit of the spider Scytodes thoracica.

This article deals with the analysis of the contact complementarity problem for Lagrangian systems subjected to unilateral constraints, and with a singular mass matrix and redundant constraints. Previous results by the authors on existence and uniqueness of solutions of some classes of variational inequalities are used to characterize the well-pose...

This paper deals with the well-posedness of a class of multivalued Lur’e systems, which consist of a nonlinear dynamical system in negative feedback interconnection with a static multivalued nonlinearity. The objective is to provide a detailed analysis of the conditions which guarantee that a certain operator, constructed from the static nonlineari...

We present in this paper a mathematical model related with calcium oscillations in a cardiac mitochondria. The model consists of a system of partial differential equations stated in mitochondrial modules. A finite element method for numerical approximation of the problemis applied; and numerical simulations showing steady state and transient behavi...

Using tools from set-valued and variational analysis, we propose a mathematical formulation for a power DC-DC Buck converter. We prove the existence of trajectories for the model. A stability and asymptotic stability results are established. The theoretical results are supported by some numerical simulations with a discussion about explicit and imp...

We study the evolution of a class of quasistatic problems, which describe frictional contact between a body and a foundation. The constitutive law of the materials is elastic, or visco-elastic: with short or long memory, and the contact is modelled by a general subdifferential condition on the velocity. We derive weak formulations for the models an...

In this paper, we present a new and alternative mathematical modelling and analysis methodology for a class of non-regular electronic circuits through a rectifier-stabilizer circuit using the superpotentiel of Moreau and the theory of variational inequalities. This methodology is suitable for engineers to study a large class of applications.

The main object of this paper is to present a general mathematical theory applicable to the study of a large class of linear variational inequalities arising in electronics. Our approach uses recession tools so as to define a new class of problems that we call “semi-complementarity problems”. Then we show that the study of semi-complementarity prob...

This paper analyzes the existence and uniqueness issues in a class of multivalued Lur’e systems, where the multivalued part is represented as the subdifferential of some convex, proper, lower semicontinuous function. Through suitable transformations the system is recast into the framework of dynamic variational inequalities and the well-posedness (...

In this paper, we develop a new approach to study a class of nonlinear generalized ordered complementarity problems.

This paper focuses on a part of the presentation given by the third author at the Shanghai Forum on Industrial and Applied Mathematics (Shanghai 2006). It is related to the existence of a periodic solution of evolution variational inequalities. The approach is based on the method of guiding functions.

The main object of this paper is to present an existence and uniqueness result for a class of variational inequalities which is of particular interest to study electrical circuits involving devices like transistors.

Using the recession analysis we study necessary and sufficient conditions for the existence and the stability of a finite semi-coercive variational inequality with respect to data perturbation. Some applications of the abstract results in mechanics and in electronic circuits involving devices like ideal diode and practical diode are discussed.

In this paper we present an extension of Moreau’s sweeping process for higher order systems. The dynamical framework is carefully introduced, qualitative, dissipativity, stability, existence, regularity and uniqueness results are given. The time-discretization of these nonsmooth systems with a time-stepping algorithm is also presented. This differe...

Keywords See also References

In this paper, we study a large class of variational inequalities that can be used to deal with some highly nonlinear phenomena in electronics like switching and clipping. Using recession tools like recession cones and recession functions, we first introduce a new class of problem that we call a ”semi- complementarity problem”. Then we show that th...

Keywords See also References

In this paper, using the Brouwer topological degree, the authors prove an existence result for finite variational inequalities. This approach is also used to obtain the existence of periodic solutions for a class of evolution variational inequalities.

In this paper, we develop a mathematical approach which can be used to display, in a systematic, rigorous and efficient way, various qualitative properties of a large class of dynamical and quasi-steady biochemical models. The applicability of the methodology has been examined for various biochemical reactions, enzyme kinetics and multi enzyme syst...

In this paper, using the Brouwer topological degree, we prove an existence result for finite variational inequalities. This ap- proach is also applied to obtain the existence of periodic solutions for a class of evolution variational inequalities. 2000 Mathematics Subject Classification 49540,49520,35K85

The aim of this paper is the study of semicoercive variational hemivariational inequalities. For this study the critical point theory of Ambrosetti, Rabinowitz and Szulkin has been extended for nonsmooth functionals. Moreover, a Saddle Point Theorem and a symmetric version of the Mountain Pass Theorem have been used. After the existence proof of th...

In this paper, we show how the approach of Moreau and Panagiotopoulos can be used to develop a suitable method for the formulation and mathematical analysis of circuits involving devices like diodes and thyristors.

The present paper concerns the hemivariational inequality approach to the laminated plate theory under abstract subdifferential conditions. The mechanical problem is formulated as a nonlinear eigenvalue problem for hemivariational inequalities and a corresponding bifurcation theory is given. Die vorliegende Arbeit behandelt den hemivariationalen Un...

A guiding function method for a class of variational inequalities is developed.

In this paper, the superpotential approach of Moreau is used in order to propose a rigorous mathematical formulation of a nonsmooth spatial model of vehicle dynamics. A 18 degrees of freedom mathematical model of three dimensional vehicle dynamics is presented in the paper considering rigid body dynamics, dynamics of tire suspensions as well as tir...

A simple approach and an algorithm are proposed to solve the quasistatic rolling frictional contact problem between an elastic cylinder and a fiat rigid body. The discretization is based on the boundary element method. The unilateral frictional contact problem (nonsmooth but monotone) is formulated in a compact form as a nonsymmetric linear complem...

In this paper, we develop a mathematical tool that can be used to state necessary conditions of asymptotic stability of isolated stationary solutions of a class of unilateral dynamical systems. More precisely, nonlinear evolution variational inequalities are considered. Instability criteria are also given. Applications can be found in mechanics or...

This paper is devoted to the study of the extension of the invariance lemma to a class of hybrid dynamical systems, namely evolution variational inequalities. Applications can be found in models of electrical circuits with ideal diodes or oligopolistic market equilibrium.

We present existence results for general variational inequalities without monotonicity or coercivity assumptions. It relies on a Leray-Schauder degree approach and provides additional information about the location of solutions.

This paper deals with the characterization of the stability and instability matrices for a class of unilaterally constrained dynamical systems, represented as linear evolution variational inequalities (LEVI). Such systems can also be seen as a sort of differential inclusion, or (in special cases) as linear complementarity systems, which in turn are...

A LaSalle's Invariance Theory for a class of first-order evolution variational inequalities is developed. Using this approach, stability and asymptotic properties of important classes of second-order dynamic systems are studied. The theoretical results of the paper are supported by examples in nonsmooth Mechanics and some numerical simulations.Résu...

This paper deals with the characterisation of the stability and unstability matrices for a class of unilaterally constrained dynamical systems, represented as linear evolution variational inequalities (LEVI). Examples show that the stability of the unconstrained system and that of the constrained system, may drastically differ. Various criteria are...

This paper deals with the characterisation of the stability and unstability matrices for a class of unilaterally constrained dynamical systems, represente- d as linear evolution variational inequalities (LEVI). Such systems can also be seen as a sort of differential inclusion, or (in special cases) as linear complementarity systems, which in turn a...

The aim of this Chapter is devoted to the study of certain classes of unilateral eigenvalue problems, i.e. eigenvalue problems for variational inequalities and hemivariational inequalities. The study of unilateral eigenvalue problems has been originated by Benci and Micheletti [19], Benci [20], Beira da Veiga [17], Do [52], [53], Kucera, Necas and...

The minimax methods developed in Chapter 4 are here used to study the existence and multiplicity of periodic solutions and homoclinic trajectories for nonsmooth Hamiltonian systems corresponding to oscillator models of the types considered in Section 2.11.10. In writing this Chapter we have primarily followed the works of Adly and Goeleven [2], [3]...

The aim of this Section is to discuss through some pilot models corresponding to hyperbolic inequality problems the maximal monotone approach, the Galerkin method and the Minimax method respectively. This Chapter relies primarily on the works of Bernardi and Pozzi [22], Brézis [29], Goeleven, Miettinen and Panagiotopoulos [90], Goeleven and Motrean...

In this Chapter we study some pilot models of parabolic variational and hemivariational inequalities. The Chapter is primarily based on the works of Brézis [29], Goeleven and Motreanu [79], [91], Miettinen [125] and Quittner [156].

The aim of this Section is to discuss in details the solutions of inequality problems of the form: $$\begin{array}{*{20}{c}} {u \in C,} \\ {\left\langle {Au - f,v - u} \right\rangle + \Phi \left( v \right) - \Phi \left( u \right)} \\ { + \int {_Tj_y^0\left( {x,u\left( x \right);v\left( x \right) - u\left( x \right)} \right)d\mu \geqslant 0,\forall...

In this Chapter we explain the origins of Unilateral Mechanics and of the Inequality Problems. To do this we use the two notions of convex and of nonconvex superpotentials. We consider boundary conditions resulting from convex or nonconvex, nonsmooth energy functions using the concept of subdifferential or of generalized gradient studied in Chapter...

We know from Chapter 2 that, if we intend to consider concrete problems in unilateral Mechanics involving both monotone and nonmonotone unilateral boundary (or interior) conditions, then we have in general to deal with a nonsmooth and nonconvex energy functional — expressed as the sum of a locally Lipschitz function \(\Phi :X \to \mathbb{R}\) and a...

The first purpose of this Chapter is to list and prove the fundamental existence theorems applicable to the study of inequality problems. Variational and hemivariational inequalities are studied for several important classes of operators among which monotone and hemicontinuous operators, semicoercive operators, nonlinear perturbations of semicoerci...

The purpose of this chapter is to provide some notions and fundamental results of convex analysis which will be used throughout this book. Starting with the notion of convexity, some basic results on convex and lower semi-continuous functionals are given. Particular attention is paid to the separation theorems of convex sets. There follow some resu...

The aim of this Chapter is to discuss an approach based on the use of topological tools (Leray-Schauder degree and continuation results) in a way that is suitable in the setting of unilateral analysis. Particular attention is paid to some nice and fundamental theorems. The material developed in this Chapter will be used later in various directions....

The aim of this paper is to discuss the mathematical strategies permitting the treatment of second order unilateral systems involving singular mass, damping, and stiffness matrices. Reduction methods are used here to transform second order differential inclusions in first order ones and classical results on differential inclusions are considered in...

A rock’s dynamic contact model taking into account friction and adhesion phenomena is discussed. It consists of a hemivariational inequality because of the adhesion process. A weak solution is obtained as a limit of a sequence of solutions to some regularized problems after establishing the necessary estimates.KeywordsFrictional ContactSurface Aspe...

The paper presents existence results for solutions to a nonsmooth hyperbolic problem in the form of a hemivariational inequality separately in the nonresonant and resonant cases.

In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a Sig...

We present a model for a nonlinear mechanical device which has a negative spring constant over a portion of its displacement range. It consists of a vertically compressed spring attached to a mass which is restricted to move on a horizontal rail. The motion is accompanied with friction, which is modeled by the Coulomb law. The model is formulated a...

Dynamic frictional contact with adhesion of a viscoelastic body and a foundation is formulated as a hemivariational inequality. This may model the dynamics of rock layers. The normal stress–displacement relation on the contact boundary is nonmonotone and nonconvex because of the adhesion process. A sequence of regularized problems is considered, th...

We describe and analyze a frictional problem for a system with a compressed spring which behaves as if it has a spring constant that is negative over a part of its extension range. As a result, the problem has three critical points. The friction is modeled by the Coulomb law. We show that there are three separate stick regions for some values of th...

The aim of this paper is to discuss a mathematical solution procedure to solve a Ramsay-type growth model that explains the fundamentals of consumption and capital accumula-tion in a dynamic equilibrium setting. The problem is formulated as a system of recursive equations and studied through some numerical experiments for the time path of the diffe...

The aim of this paper is to deduce general conditions on a matrix M and a closed convex set K ensuring the solvability for each q∈ℝ N of the linear variational inequality LVI(M,q,K): Find u∈K such that (Mu) T (v-u)≥q T (v-u),∀v∈K· Using the Brouwer degree we reduce the study of the original problem to the one of relaxed LCPs defined on the recessio...

Dynamic hemivariational inequalities are studied in the present paper. Starting from their solution in the distributional sense, we give certain existence and approximation results by using the Faedo–Galerkin method and certain compactness arguments. Applications from mechanics (viscoelasticity) illustrate the theory.

In the present paper a method is presented for the analysis of rock interface. The mechanical response of rock interface involves nonlinear unilateral phenomena as debonding and slip along the interface. Under the usual simplifying assumption of shear-normal stress decoupling, the problem has been studied in [1,2] by means of variational and hemiva...

The mathematical modeling of engineering structures containing members capable of transmitting only certain type of stresses or subjected to noninterpenetration conditions along their boundaries leads generally to variational inequalities of the form , where C is a closed convex set of (kinematically admissible set), (loading strain vector), and...

This paper contains some existence and multiplicity results for periodic solutions of second order nonautonomous and nonsmooth Hamiltonian systems involving nonconvex superpotentials. This study is achieved by proving the existence of homoclinic solutions. These solutions are constructed as critical points of the corresponding nonconvex and nonsmoo...

The aim of this paper is to extend the results obtained in hyperelasticity by Ball 8,9, Ciarlet and Neĉas 17 and Gugat 30, 31 by considering a large class of nonmonotone possibly multivalued boundary conditions. Thus we are led to noncoercive hemivariational inequalities related to hyperelastic polyconvex materials

The present chapter deals with multiplicity results for the eigenvalue problems of hemivariational inequalities. First we give a general minimax approach which permits the use of the corresponding linear eigenvalue problem for the determination of eigenvectors of a hemivariational inequality. Then the case of even nonconvex superpotential j(x, ·) i...

In this paper we prove firstly that if f:X→ℝ is a locally Lipschitz function, bounded from below and invariant to a discrete group of dimension N is a suitable sense, acting on a Banach space X, then the problem: find u∈X such that o∈∂ f(u) (here ∂f(u) denotes Clarke's generalized gradient of f at x) admits at least N+1 orbits of solutions. Then, f...

A soil mercury survey was conducted near the town of Calistoga, California to identify and delineate a buried fault system that is thought to control the upwelling of low-to-moderate temperature geothermal fluids in the upper Napa Valley. Soil samples were collected at 100 m intervals along traverses that crossed hot springs and existing geothermal...

This paper concerns the study of superpotential laws in analytical mechanics which may be derived from generally non-everywhere differentiable energy functions through subdifferentiation. A generalization of d'Alembert's principle is proposed dealing with problems involving unilateral constraints.

The aim of this paper is to study a class of eigenvalue problems of variational-hemivariational type.

The aim of this paper is to describe the aspects related to the mathematical formulation of the unilateral contact problem including rigid body motions in structural mechanics. Both Finite Element and Boundary Element formulations are considered.

The study of the equilibrium of an object-robotic hand system including nonmonotone adhesive effects and nonclassical friction effects leads to new inequality methods in robotics. The aim of this paper is to describe these inequality methods and provide a corresponding suitable mathematical theory.

A hemivariational inequality model for adhesive grasping problems is proposed and studied in this paper. The unilateral frictionless and frictional contact effects between the fingertips and the grasping object that lead to linear complementarity problems with singular matrices for the study of static equilibrium of the gripper-object system are ge...

A hemivariational inequality model for adhesive grasping problems is proposed and studied in this paper. The unilateral frictionless and frictional contact effects between the fingertips and the grasping object that lead to linear complementarity problems with singular matrices for the study of static equilibrium of the gripper-object system are ge...

The hemivariational inequality approach is used in order to establish the existence of solutions to a large class of noncoercive constrained problems in a reflexive Banach space, in which the set of all admissible elements is not convex but fulfills some star-shaped property.

In this paper, we prove a uniqueness theorem for a class of complementarity problems involving M-matrices.

The hemivariational inequality approach of P.D. Panagiotopoulos [18]-[22] is used so as to establish the existence of solutions to a class of noncoercive and nonconvex constrained problems in a reflexive Banach space.

This paper studies the solvability of a general class of variational inequalities. Existence of periodic solutions for noncoercive variational inequalities will be proved.

Using a new compact imbedding theorem of C. De Coster and M.Willem [5], we prove the existence of homoclinic orbits for a class of hemivariational inequalities.

Necessary conditions for the stability of elastic bodies subjected to nonmonotone multivalued boundary conditions are derived. These conditions are assumed to be derived from nonconvex and nonsmooth, quasidifferentiable energy functions. A ‘difference convex’ approximation of the potential energy function is written based on an appropriate quasidif...