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[B]: Books, [P]: Preprints, [J]: Journal articles, [BC]: Book Chapters,
[CP]: Conference Proceedings.
Books
[B1] 
S. Adly.
A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics,
SpringerBriefs in Mathematics (2019).
About this book.
Abstract
Abstract:
This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational
inequalities, complementarity systems, differential inclusions, secondorder dynamics, Lur'e systems and Moreau's sweeping
process.
The field of nonsmooth dynamics is of great interest to mathematicians, mechanicians, automatic controllers and engineers. The
present volume acknowledges this transversality and provides a multidisciplinary view as it outlines fundamental results in
nonsmooth dynamics and explains how to use them to study various problems in engineering. In particular, the author explores the
question of how to redefine the notion of dynamical systems in light of modern variational and nonsmooth analysis.
With the aim of bridging between the communities of applied mathematicians, engineers and researchers in control theory and
nonlinear systems, this brief outlines both relevant mathematical proofs and models in unilateral mechanics and electronics.

Submitted Papers
[P8] 
S. Adly and Manh Hung Le.
Solving Inverse Pareto Eigenvalue Problems,
Submitted on February 2022 .
Abstract
Abstract:
We compare in this note a variety of methods for solving inverse Pareto eigenvalue problems which are aimed at constructing matrices
whose Pareto spectrum contains a prescribed set of distinct reals. We choose to deal with such problems by first formulating them as nonlinear systems of
equations which can be smooth or nonsmooth, depending on the chosen approach, and subsequently adopt Newton type methods to solve the corresponding systems.
Our smooth approach includes the Squaring Trick (ST) and the socalled Mehrotra Predictor Corrector Method (MPCM), adapted in this context to inverse Pareto eigenvalue
complementarity problems. For the nonsmooth approach, we consider the Lattice Projection Method (LPM), and two other nonsmooth methods using complementarity function
techniques, namely $\text{SNM}_{\text{FB}}$ and $\text{SNM}_{\text{min}}$ (with FischerBurmeister and minimum complementarity functions respectively).
We compare the five methods using the performance profiles (Dolan, Mor\'e), where the average number of iterations and the percentage of failures are the performance
measures. Numerical tests show that among the methods considered, $\text{SNM}_{\text{FB}}$ performs the best in terms of the number of failures whereas LPM surpasses
all other methods with respect to the number of iterations. Finally, we point out possible extensions of the discussed methods to the inverse quadratic pencil eigenvalue
complementarity problem.

[P7] 
S. Adly and H. Attouch.
On the limit form of the SuBoydCandès inertial dynamic version of Nesterov's method when the viscous parameter becomes large,
Submitted on February 2022 .
Abstract
Abstract:
In a Hilbert setting, our study concerns the dynamical system introduced by SuBoydCand\`es as a low resolution ODE of Nesterov's accelerated gradient method and of the Ravine method.
This inertial system ${\rm (AVD)}_{\alpha}$ is driven by the gradient of the function $f$ to be minimized, and is damped by a viscous friction whose coefficient is equal to $\alpha/t$, and therefore tends to zero when $t \to + \infty$.
Taking $\alpha$ sufficiently large plays a crucial role in the asymptotic convergence properties of the trajectories generated by this sytem.
In particular, for a general convex function $f$, the condition $\alpha >3$ guarantees the asymptotic convergence rate of the values $o \left( 1/t^2 \right)$, as well as the convergence of the trajectories towards optimal solutions.
On the basis of temporal scaling techniques, we obtain, for $ \alpha $ large, equivalent simple continuous dynamics which are autonomous and of the first order in time.
Precisely, we show that a judicious time rescaling of ${\rm (AVD)}_{\alpha}$ results in trajectories close to those of the continuous steepest descent associated with $f$. This explains the passage from
$1/t$ to $1/t^2$ of the convergence rate of values between the steepest descent method and the Nesterov accelerated gradient method (as well as the Ravine
accelerated gradient method). At the same time, new results are obtained concerning the complexity analysis of the Nesterov method and of the Ravine method.
Some numerical experiments are provided to verify the theoretical results.

[P6] 
S. Adly, Nguyen Nang Thieu and Nguyen Dong Yen.
On the limit form of the SuBoydCandès inertial dynamic version of Nesterov's method when the viscous parameter becomes large,
Submitted on February 2022 .
Abstract
Abstract:
In this paper, we study some classes of sweeping processes with velocity constraints in the moving set. In addition to the solution existence and the
solution uniqueness for the case of a moving convex constraint set, some results on the solution existence and the solution multiplicity where the
constraint set is a finite union of disjoint convex sets are also obtained. Our main tool is a theorem on the solution sensitivity of parametric variational
inequalities. Beside the traditional requirement that the constraint set moves continuously in the Hausdorff distance sense, we intensively use a new assumption
on the local Lipschitzlikeness of the constraint setvalued mapping. The obtained results are compared with the existing ones and analyzed by several examples.

[P5] 
S. Adly, H. Attouch and Manh Hung Le.
Firstorder inertial optimization algorithms with threshold effects associated with dry friction,
Submitted on July 2021 .
Abstract
Abstract:
In a Hilbert space setting, we consider a new first order optimization algorithm which is obtained by temporal discretization of a damped inertial dynamic involving
dry friction. The function $f$ to be minimized is assumed to be differentiable (not necessarily convex).
The dry friction potential function $\phi$, which has a sharp minimum at the origin, enters the algorithm via its proximal mapping, which acts as a soft
thresholding operator on the sum of the velocity and the gradient terms. After a finite number of steps, the structure of the algorithm changes, losing its
inertial character to become the steepest descent method. The geometric damping driven by the Hessian of $f$ makes it possible to control and attenuate the
oscillations.
The algorithm generates convergent sequences when $f$ is convex, and in the nonconvex case when $f$ satisfies the KurdykaLojasiewicz property. As a remarkable
property, the convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence towards zero. The study is then
extended to the case of a nonsmooth convex function $f$, in which case the algorithm involves the proximal operators of $f$ and $\phi$ separately.
Then, applications are given to the Lasso problem and nonsmooth d.c. programming.

[P4] 
S. Adly, M. Haddou and Manh Hung Le.
Interior point methods for solving coneconstrained eigenvalue problems,
Submitted on March 2021.
Abstract
Abstract:
In this paper, we propose to solve coneconstrained eigenvalue problems by using interiorpoint methods. Precisely, we focus the study on an adaptation of the Mehrotra
Predictor Corrector Method (MPCM) and a NonParametric Interior Point Method (NPIPM).
We compare these two methods with the Lattice Projection Method (LPM) and the SoftMax Method (SM).
The performance profiles, on a set of data generated from the MatrixMarket, highlight the efficiency of MPCM and NPIPM for solving eigenvalue complementarity problems.
We also consider an application to a concrete and large size situation corresponding to a geomechanical fracture problem.
Finally, we discuss the extension of MPCM and NPIPM methods to solve quadratic pencil eigenvalue problems under conic constraints.

[P3] 
S. Adly, L. Bourdin and G. Dhar.
The application of a universal separating vector lemma to optimal sampleddata control problems with nonsmooth Mayer cost function,
Submitted on March 2021.
Abstract
Abstract:
In this paper we provide a Pontryagin maximum principle for optimal sampleddata control problems with nonsmooth Mayer cost function. Our investigation leads us
to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of
a universal separating vector which belongs to the convex envelope of a given set of separating vectors of the singletons of a given compact convex set.
This socalled universal separating vector lemma is used, together with packages of convex control perturbations, to derive a Pontryagin maximum principle
for optimal sampleddata control problems with nonsmooth Mayer cost function. As an illustrative application of our main result we solve a simple example by
implementing an indirect numerical method.

[P2] 
S. Adly, L. Bourdin and F. Caubet.
The derivative of a parameterized mechanical contact problem with a Tresca's friction law involves Signorini unilateral conditions,
Submitted on November 2019.
Abstract
Abstract:
The present paper investigates the sensitivity analysis, with respect to righthand source term perturbations, of a mechanical
contact problem involving a Tresca's friction law. The weak formulation of this problem leads to a variational inequality of
the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using
the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex
subdifferential of the friction functional on the Sobolev space~$\H^1(\Omega)$ and show that all its subgradients satisfy a
PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the
twice epidifferentiability, concept introduced and thoroughly studied by R.T. Rockafellar, we show the differentiability
of the parameterized Tresca's solution and that its derivative satisfies Signorini unilateral conditions. Some numerical
simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first
time that the concept of twice epidifferentiability is applied in the context of mechanical contact problems, which makes this
contribution new and original in the literature.

[P1] 
S. Adly, D. Goeleven and R. Oujja.
Wellposedness of nonsmooth Lurie dynamical systems involving maximal monotone operators,
Submitted on June 2020.
Abstract
Abstract:
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 Postinikov \cite{lur'e} in the early 1940s, constitutes an important tool in
the theory of control systems. Lurie dynamical systems have been studied extensively in the literature with nonlinear (but smooth) feedback functions, that can be
formulated as an ordinary differential equation. Many concrete applications in engineering can be modeled by a setvalued feedback law in order to take into account
the nonsmooth relation between the output and the state variables. In this paper, we show the wellposedness of nonsmmoth Lurie dynamical systems involving maximal
monotone operators. This includes the case where the setvalued law is given by the subdifferential of a convex, proper and lower semicontinuous function.
Some existence and uniqueness results are given depending on the data of the problem and particularly the interplay between the matrix $D$ and the setvalued
map $\F$. We will also give some conditions ensuring that the the extended resolvent $(D+\F)^{1}$ is single valued and Lipschitz continuous. The main tools used
are derived from convex analysis and maximal monotone theory.

Refereed Publications
2022
DennisMoré condition for setvalued vector fields and the superlinear convergence of Broyden updates in Riemannian manifolds,
(with Huynh Van Ngai and Van Vu Nguyen)
Journal of Convex Analysis 29, No. 3, (2022).
Abstract
Abstract:
This paper deals with the quasiNewton type scheme for solving generalized equations involving pointtoset vector fields on Riemannian manifolds. We establish some conditions
ensuring the superlinear convergence for the iterative sequence which approximates a solution of the generalized equations. Such conditions can be viewed as an extension of the classical
DennisMoré theorem in [1] as well as the Riemannian DennisMoré condition established in the work [2]. Furthermore, we also apply these results to consider the convergence of a
Broydentype update for the problem of solving generalized equations in Riemannian context. Our results are new even for classical equations defined by singlevalued vector fields.
[1] J. E. Dennis and J.J. Moré, A characterization of
superlinear convergence and its application to quasiNewton methods,
Mathematics of Computation, 28 (1974), pp. 549560.
[2] K.A. Gallivan, C.Qi, and P.A. Absil}, HighPerformance Scientific
Computing: Algorithms and Applications, Springer London, 2012, ch. A
Riemannian DennisMoré Condition, pp.281293.
2021
Sensitivity analysis of maximal monotone inclusions via the protodifferentiability of the resolvent operator,
(with R.T. Rockafellar).
Mathematical Programming no. 12, Ser. B, 37–54 (2021).
Abstract
Abstract:
This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximal
monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account.
Using the concept of protodifferentiability of a multifunction and the notion of semidifferentiability of a singlevalued map,
we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the
protoderivative of the resolvent operator associated to the maximal monotone parameterized variational inclusion.
This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being
itself a solution of a variational inclusion involving the semiderivative and the protoderivative of the associated maps.
An application to the study of the sensitivity analysis of a parametrized primaldual composite monotone inclusion is given.
Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their
derivatives belong to the derivative of the associated KuhnTucker set.
Asymptotic behavior of Newtonlike inertial dynamics involving the sum of potential and nonpotential terms,
(with Hedy Attouch and Van Nam Vo).
Journal of Fixed Point Theory and Applications, Paper No. 17, 30 pp (2021).
Abstract
Abstract:
In a Hilbert space $\HH$, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations
involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator $A= \nabla f +B $
where $\nabla f$ is the gradient of a continuously differentiable convex function $f$, and $B$ is a nonpotential monotone and
cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled
respectively by the Hessian of the potential $f$ and by a Newtontype correction term attached to $B$. Based on a fixed point
argument, we show the wellposedness of the Cauchy problem. Then we show the weak convergence as $t\to+\infty$ of the
generated trajectories towards the zeros of $\nabla f +B$. The convergence analysis is based on the appropriate setting of
the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and
attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the secondorder evolution equation as
a firstorder dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open
the door to the design of new firstorder accelerated algorithms in optimization taking into account the specific properties of
potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of
the nonpotential term.
New metric properties for proxregular sets,
(with Florent Nacry and Lionel Thibault).
Mathematical Programming 189 (2021), no. 12, Ser. B, 7–36.
Abstract
Abstract:
In this paper, we present diverse new metric properties that proxregular sets shared with convex ones. At the heart of our work
lie the LegendreFenchel transform and complements of balls. First, we show that a connected proxregular set is completely
determined by the LegendreFenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a
function is also the right tool to extend, to the context of proxregular sets, the famous connection between the distance
function and the support function of a convex set. On the other hand, given a proxregular set, we examine the intersection of
complements of open balls containing the set.
We establish that the distance of a point to a proxregular set is the maximum of the distances of the point from boundaries of
all such complements separating the set and the point. This is in the line of the known result expressing the distance from a
convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show
that the class of proxregular sets have good properties known in convex analysis.
DouglasRachford splitting algorithm for solving statedependent maximal monotone inclusions,
(with Le Ba Khiet).
Optimization Letters, 15, no. 8, 28612878 (2021).
Abstract
Abstract:
In this paper, we provide a new application of DouglasRachford splitting method for finding a zero of two maximal monotone
operators where one of the operators depends on the state. Our proposed algorithms are simple with significant rate of
convergence and can be implemented under general conditions. Applications to generalized Nash games and quasivariational
inequalities are provided to illustrate the obtained results.
Finite time stabilization of continuous inertial dynamics combining dry friction with Hessiandriven damping,
(with Hedy Attouch).
Journal of Convex Analysis, Vol. 28, no. 2, 281310, (2021).
Abstract
Abstract:
In a Hilbert space $\H$, we study the stabilization in finite time of the trajectories generated by a continuous (in time $t$)
damped inertial dynamic system. The potential function $f:\cH \to \R$ to be minimized is supposed to be differentiable, not necessarily convex.
It enters the dynamic via its gradient. The damping results from the joint action of dry friction, viscous friction, and a geometric damping driven by
the Hessian of $f$.
The dry friction damping function
$\phi:\H \to \R_+$, which is convex with a sharp minimum at the origin (typically $\phi(x) = r \x\$ with $r>0$), enters the dynamic via its subdifferential.
It acts as a soft threshold operator on the velocities, and is at the origin of the stabilization property in finite time.
The Hessian driven damping, which enters the dynamics in the form $\nabla^2 f (x(t)) \dot{x} (t)$,
permits to control and attenuate the oscillations which occur naturally with the inertial effect.
We give two different proofs, in a finite dimensional setting, of the existence of strong solutions
of this secondorder differential inclusion. One is based on a fixed point argument and the use of LeraySchauder theorem, the other one is
based on the Yosida approximation technique and the Mosco convergence. We also give an existence and uniqueness result in a general Hilbert
framework by assuming that the hessian of the function $f$ is Lipschitz continuous on the bounded sets of $\cH$.
Then, we study the convergence properties of the trajectories as $t \to +\infty$, and show their stabilization property in finite time.
The convergence results tolerate the presence of perturbations, errors, under the sole assumption of their asymptotic convergence to zero.
Then, we extend the study to the case of a nonsmooth convex function $f$.
2020
Finite convergence of proximalgradient inertial algorithms combining dry friction with Hessiandriven damping,
(with Hedy Attouch).
SIAM Journal on Optimization (SIOPT), Vol. 30, No. 3, pp. 21342162 (2020).
Abstract
Abstract:
In a Hilbert space $H$, we introduce a new class of proximalgradient algorithms
with finite convergence properties.
These algorithms naturally occur as discrete temporal versions of an inertial differential inclusion which is damped under the joint action of three dampings: a viscous damping, a geometric damping driven by the Hessian and a dry friction damping.
The function $f:H \to R$ to be minimized is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient.
The dry friction damping function
$\phi:H \to \R_+$ is convex with a sharp minimum at the origin, (typically $\phi(x) = r \x\$ with $r>0$). It enters the algorithm via its proximal mapping, which acts as a soft threshold operator on the velocities.
The geometrical damping driven by the Hessian intervenes in the dynamics in the form $\nabla^2 f (x(t)) \dot{x} (t)$. By treating this term as the time derivative of $ \nabla f (x (t)) $, this gives, in discretized form, firstorder algorithms.
The Hessian driven damping allows to control and to attenuate the oscillations. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero.
Replacing the potential function $f$ by its Moreau envelope, we extend the results to the case of a nonsmooth convex function $f$. In this case, the algorithm involves the proximal operators of $f$ and $\phi$ separately.
Several variants of this algorithm are considered, including the case of the Nesterov accelerated gradient method. We then consider the extension in the case of additive composite optimization, thus leading to splitting methods. Numerical experiments are given for Lassotype problems. The performance profiles, as a comparison tool, highlight the effectiveness of a variant of the Nesterov accelerated method with dry friction and Hessiandriven damping.
A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction,
(with Daniel Goeleven)
Evolution Equations and Control Theory, no. 4, 915–934. (2020).
Abstract
Abstract:
In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the
modelling of a rotary oil drilling system with friction. We study different kinds of frictions and analyse the mathematical
properties of the involved dynamical systems. We show that using a general Stribeck model for the frictional contact, we can
formulate the rotary drilling system as a wellposed evolution variational inequality. Several numerical simulations are also
given to illustrate both the model and the theoretical results.
On Evolution quasiVariational Inequalities and Implicit statedependent Sweeping Processes,
(with T. Haddad)
Discrete and Continuous Dynamical SystemsS, 13, no. 6, 17911801, (2020).
Abstract
Abstract:
In this paper, we study a variant of the statedependent sweeping process with velocity constraint.
The constraint $C(\cdot,u)$ depends upon the unknown state $u$, which
causes one of the main difficulties in the mathematical treatment of
quasivariational inequalities. Our aim is to show how a fixed point
approach can lead to an existence theorem for this implicit
differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness
theorem in the case where the moving set $C$ does not depend explicitly on the state $u$ (i.e. $C:=C(t)$) given in \cite{AHT},
we prove a new existence result of solutions of the quasivariational sweeping process in the infinite dimensional Hilbert spaces
with a velocity constraint. Contrary to the classical statedependent sweeping process, no conditions on the size of the Lispchitz constant of
the moving set, with respect to the state, is required.
2019
A coderivative approach to the robust stability of composite parametric variational systems. Applications in nonsmooth mechanics.
Journal of Optimization Theory and Applications, 180, no. 1, pp 6290, (2019).
Abstract
Abstract:
The main concern of this paper is to investigate the Lipschitzianlike stability property (namely Aubin property) of the solution map of
possibly nonmonotone variational systems with composite superpotentials. Using Mordukhovich coderivative criterion and a secondorder
subdifferential analysis, we provide simple and verifiable characterizations of this property in terms of the data involved in the problem.
Applications are given in nonsmooth mechanics.
Lyapunov stability of differential inclusions with Lipschitz Cusco perturbations of maximal
monotone operators,
(with A. Hantoute and Bao Tran Nguyen).
SetValued and Variational Analysis, 28, no. 2, 345368, (2020).
Abstract
Abstract:
We give criteria for weak and strong invariant closed sets for differential
inclusions given in $\mathbb{R}^{n}$ and governed by Lipschitz Cusco
perturbations of maximal monotone operators. Correspondingly, we provide
different characterizations for the\ associated strong Lyapunov functions.
The resulting conditions only depend on the data of the system.
An extended conjugate duality for generalized semiinfinite programming problems via a convex decomposition,
(with A. Aboussoror and S. Salim)
Optimization, 69, no. 78, 16351654, (2020).
Abstract
Abstract:
We present an extended conjugate duality for a generalized
semiinfinite programming problem $({\mathcal P})$. The extended
duality is defined in the context of the absence of convexity of
problem
$({\mathcal P})$, by means of a decomposition into a family of
convex subproblems and a conjugate dualization of the
subproblems. Under appropriate assumptions, we establish strong extended
duality and provide necessary and sufficient optimality conditions
for problem $({\mathcal P})$. These extended conjugate duality and
optimality conditions are new in the literature of generalized
semiinfinite programming.
On a decomposition formula for the resolvent operator of the sum of two setvalued maps with monotonicity assumptions,
(with L. Bourdin).
Applied Mathematics and Optimization, 80, no. 3, 715–732, (2019).
Abstract
Abstract:
The aim of the present work is to provide an explicit decomposition formula for the resolvent operator~$\J_{A+B}$ of the sum
of two setvalued maps~$A$ and~$B$ in a Hilbert space. For this purpose we introduce a new operator, called the~$A$resolvent
operator of~$B$ and denoted by~$\JAB$, which generalizes the usual notion. Then, our main result lies in the decomposition
formula~$\J_{A+B}=\JA \circ \JAB$ holding true when~$A$ is monotone. Several properties of~$\JAB$ are deeply investigated in
this paper. In particular the relationship between~$\JAB$ and an extension of the classical DouglasRachford operator is
established, which allows us to propose a weakly convergent algorithm that computes numerically~$\JAB$ (and thus~$\J_{A+B}$
from the decomposition formula) when~$A$ and~$B$ are maximal monotone. In order to illustrate our theoretical results, we give
an application in elliptic PDE's. Precisely the decomposition formula is used to point out the relationship between the classical
obstacle problem and a new nonlinear PDE involving a partially blinded elliptic operator. Some numerical experiments, using
finite element method, are carried out in order to support our approach.
Optimality Conditions for Strong Semivectorial Bilevel Programming Problems via a Conjugate Duality,
(with A. Aboussoror and F.E. Saissi)
Journal of Pure and Applied Functional Analysis, 4, no. 2, 151–176, (2019).
Abstract
Abstract:
We are concerned with a strong semivectorial
nonlinear bilevel programming problem where the upper and lower
levels are vectorial and scalar respectively. For such a problem
we give a duality approach via scalarization, regularization and a
conjugate duality. Then, via this duality approach, we provide
necessary
and sufficient optimality conditions for the initial semivectorial bilevel programming problem. This duality approach extends the one given in [1] from the scalar case to the semivectorial one.
[1] A. Aboussoror, S. Adly and F. E. Saissi, An extended FenchelLagrange duality approach and optimality conditions for strong bilevel programming problems, SIAM J.
Optim. Vol. 27, No. 2, pp. 12301255, 2017.
Wellposedness of discontinuous secondorder nonconvex statedependent sweeping processes,
(with F. Nacry).
Applied Mathematics and Optimization, 79 (2019), no. 2, 515546.
https://doi.org/10.1007/s0024501794469 .
Abstract
Abstract:
In this paper, we study the existence of solutions for a time and statedependent discontinuous nonconvex second order sweeping
process with a multivalued perturbation. The moving set is assumed to be proxregular, relatively ballcompact with a bounded variation. The perturbation of the normal cone is a scalarly upper semicontinuous convex valued multimapping
satisfying a linear growth condition possibly timedependent. As an application of the theoretical results, we investigate the
theory of evolution quasivariational inequalities.
Timedependent Inclusions and Sweeping Processes in Contact Mechanics,
(with M. Sofonea)
Zeitschrift fü̈r angewandte Mathematik und Physik ZAMP:
70 (2019), no. 2, Art. 39, 19 pp. https://doi.org/10.1007/s0003301910844
Abstract
Abstract:
We consider a class of timedependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result.
The proof is based on arguments of variational inequalities, convex analysis and fixed point theory.
Then we use this result to prove the unique weak solvability of a new class of Moreau's sweeping processes with constraints in velocity.
Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle.
To provide some examples we consider three viscoelastic contact problems which lead to timedependent inclusions and sweeping processes in
which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique
weak solvability of the corresponding contact problems.
Sensitivity properties of parametric nonconvex evolution inclusions with application to optimal control problems,
(with T. Zakaryan).
Setvalued and Variational Analysis, 27 (2019), no. 2, pp 549568.
http://dx.doi.org/10.1007/s112280190505z
Abstract
Abstract:
The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first and second order
involving a general class of nonconvex functions. Using recent results on the stability of the subdifferentials, with respect to
the Gamma convergence, of the associated sequence of pln (primal lower nice) and semiconvex functions,
we give some continuity properties of the solution set associated to these problems. The particular case of the parametric sweeping
process involving proxregular sets is studied in details. As an application, we study the sensitivity analysis of the generalized
Bolza/Mayer problem governed by a nonsmooth dynamic of a sweeping process type.
Statedependent implicit sweeping process in the framework of quasistatic evolution
quasivariational inequalities,
(with T. Haddad and B.K. Le).
Journal of Optimization Theory and Applications, 182, no. 2, 473493, (2019).
https://doi.org/10.1007/s109570181427x
Abstract
Abstract:
This paper deals with the existence and uniqueness of solutions for a class of statedependent sweeping processes with constrained
velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty,
we introduce a new notion called hypomonotonicitylike of the normal cone to the moving set, which is satisfied for many important cases.
Combining this latter notion with an implicit time discretization and a Cauchy technique, we obtain the strong convergence of approximate solutions
to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit
statedependent sweeping processes and quasistatic evolution quasivariational inequalities.
As an application, we study the statedependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.
Lyapunov Stability of Differential Inclusions Involving ProxRegular Sets via Maximal Monotone Operators,
(with A. Hantoute and Bao Tran Nguyen).
Journal of Optimization Theory and Applications, 182, no. 3, 906934, (2019).
https://doi.org/10.1007/s1095701814467
Abstract
Abstract:
In this paper, we study the existence and the stability in the sense of Lyapunov of solutions for differential inclusions
governed by the normal cone to a proxregular set and subject to a Lipschitzian perturbation. We prove that such, apparently, more
general nonsmooth dynamics can be indeed remodelled into the classical theory of differential inclusions involving maximal
monotone operators. This result is new in the literature and permits us to make use of the rich and abundant achievements in
this class of monotone operators to derive the desired existence result and stability analysis, as well as the continuity and
differentiability properties of the solutions. This going back and forth between these two models of differential inclusions is made
possible thanks to a viability result for maximal monotone operators. As an application, we study a Luenbergerlike observer,
which is shown to converge exponentially to the actual state when the initial value of the state's estimation remains in a
neighborhood of the initial value of the original system.
Weak Lyapunov functions and differential inclusions involving proxregular sets,
(with A. Hantoute and Bao Tran Nguyen).
Journal of Convex and Nonlinear Analysis, Volume 20, Number 1, (2019)
Abstract
Abstract:
We first give criteria for weak Lyapunov functions associated to differential inclusions, given in $\mathbb{R}^{n}$ and governed by Lipschitz
Cusco perturbations of maximal monotone operators. Next, we apply this result to study existence and stability of differential inclusions involving
normal cones to proxregular sets.
2018
Sensitivity analysis of variational inequalities via twice epidifferentiability and protodifferentiability of the proximity operator,
(with L. Bourdin).
SIAM Journal on Optimization (SIOPT), 28 (2018), no. 2, 16991725.
Abstract
Abstract:
In this paper we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of second kind in a Hilbert space.
The challenge of the present work is to take into account a perturbation on all the data of the problem. This requires special adjustments in the definitions
of the generalized first and secondorder differentiations of the involved operators and functions. Precisely, we extend the notions, introduced and thoroughly
studied by R.T.~Rockafellar, of twice epidifferentiability and protodifferentiability to the case of a parameterized lower semicontinuous convex function and
its subdifferential respectively. The link between these two notions is tied to Attouch's theorem and to the new concept, introduced in this paper, of convergent
supporting hyperplanes. The previous tools allow us to derive an exact formula of the protoderivative of the generalized proximity operator associated to a
parameterized variational inequality, and deduce the differentiability of the associated solution with respect to the parameter. Furthermore, the derivative is
shown to be the solution of a new variational inequality involving semi and twice epiderivatives of the data. An application is given to parameterized convex
optimization problems involving the sum of two convex functions (one of them being smooth). The case of smooth convex optimization problems with inequality
constraints is discussed in details. This approach seems to be new in the literature and open several perspectives towards theoretical and computational issues
in nonlinear optimization.
An implicit sweeping process approach to quasistatic evolution variational inequalities,
(with T. Haddad).
SIAM Journal on Mathematical Analysis, 50 (2018), no. 1, 761–778.
Abstract
Abstract:
In this paper, we study a new variant of the Moreau's sweeping process with
velocity constraint. Based on an adapted version of the Moreau's catchingup algorithm, we show the wellposedness (in the sense existence and
uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality.
It is wellknown that the variational
formulation of many mechanical problems with unilateral contact and friction lead to an evolution
variational inequality. As an application, we reformulate the quasistatic antiplane frictional contact
problem for linear elastic materials with short memory as an implicit sweeping process with
velocity constraint. The link between the implicit sweeping process and the quasistatic
evolution variational inequality is possible thanks to some standard
tools from convex analysis and is new in the literature.
On semicoercive sweeping process with velocity constraint,
(with B.K. Le).
Optimization letters, 12 (2018), no. 4, 831843. DOI: 10.1007/s1159001711492.
Abstract
Abstract:
In this note, by solving a variational inequality at each iteration, we study the existence of solutions for a class of
sweeping processes with velocity in the moving set, originally introduced in a recent paper [1]. Our aim is to
improve Theorem 5.1 in [1] to allow possibly unbounded moving sets. The theoretical result is supported by some examples
in nonregular electrical circuits.
[1] S. Adly, T. Haddad and L. Thibault, Convex Sweeping Process in the framework of Measure Differential Inclusions and Evolution
Variational Inequalities, Math. Program., Ser. B (2014) 148 (1), 547.
New Necessary and Sufficient Optimality Conditions for Strong Bilevel Programming Problems,
(with A. Aboussoror).
Journal of Global Optimization, 70 (2018), no. 2, 309–327.
Abstract
Abstract:
In this paper we are interested in a strong bilevel programming
problem. For such a problem, we establish necessary and
sufficient optimality conditions. Our investigation is based on
the use of a regularization of this problem and some wellknown
global optimization tools. These optimality conditions
are new in the literature and are expressed in terms of
maxmin conditions with linked constraints.
Proxregularity approach to generalized equations and image projection,
(with F. Nacry and L. Thibault).
ESAIM: Control, Optimisation and Calculus of Variations 24, pp 677708 (2018). DOI: cocv170112.
Abstract
Abstract:
In this paper, we first investigate the proxregularity behaviour of solution mappings to generalized equations. This study is realized through a nonconvex uniform RobinsonUrsescu type theorem.
Then, we derive new significant results for the preservation of proxregularity under various and usual set operations. The role and applications of proxregularity of solution sets of generalized
equations are illustrated with dynamical systems with constraints.
Secondorder State Dependent Sweeping Process with Unbounded and Nonconvex Constraints,
(with B.K. Le).
Pure and Applied Functional Analysis, no. 2, 271285 (2018).
Abstract
Abstract:
In this paper, by using an implicit discrete scheme, we prove the existence of solutions for a class of secondorder
sweeping processes with possibly unbounded and proxregular moving sets depending on both the time and the state in
a Hilbert space. We use the local excess (local onesided Lipschitz continuity property) instead of the global Hausdorff
distance to describe the way the set of constraints is moving.
Newtontype method for solving generalized equation on Riemannian manifolds,
(with H. V. Ngai and N. Van Vu)
Journal of Convex Analysis, Volume 25, No. 2, pp 341370 (2018).
Abstract
Abstract:
This paper is devoted to the study of Newtontype algorithm for solving inclusions involving setvalued maps defined
on Riemannian manifolds. We provide some sufficient conditions ensuring the existence as well as the quadratic convergence
of Newton sequence. The material studied in this paper is based on Riemannian geometry as well as variational analysis, where
metric regularity property is a key point.
Invariant sets and Lyapunov pairs for differential inclusions with
maximal monotone operators,
(with A. Hantoute and Bao Tran Nguyen).
Journal of Mathematical Analysis and Applications, Volume 457, Issue 2, 15 January 2018, pages 10171037.
Abstract
Abstract:
We give different conditions for the invariance of closed sets with respect to
differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state.
These sets are not necessarily weakly closed, while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semigroup generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for $a$Lyapunov pairs of lower semicontinuous (not necessarily weaklylsc) functions associated to these differential inclusions.
The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.
Duality and Optimality Conditions for a Class of Semivectorial
Bilevel Programming Problems,
(with A. Aboussoror and F.E. Saissi).
Vietnam Journal of Mathematics, 46 (2018), no. 1, 197–214.
Abstract
Abstract:
In this paper, we are concerned with a
semivectorial bilevel programming problem $(S)$
where the upper and lower levels are vectorial and scalar
respectively. Our aim is to give a duality approach for problem
$(S)$ via scalarization. However, the considered scalarized
problem of $(S)$ does not satisfy a constraint qualification
that we need for our duality approach. To overcome this obstacle,
we first
give a regularized problem $(S_\epsilon)$ of $(S)$ ($\epsilon>0$) whose scalarized problem $(S_\epsilon^s)$ satisfies this condition.
Afterwards, we consider the FenchelLagrange
dual to $(S_\epsilon^s)$.
Using a stability result and under
appropriate assumptions, we establish strong duality and
provide optimality conditions for $(S_\epsilon^s)$ and its dual. Then, via
the duality given for the
previous regularizedscalarized case, we provide necessary optimality conditions for a
class of properly efficient solutions of
$(S)$.
Finally, sufficient optimality conditions are given for $(S)$
without using duality. Examples are given for illustration.
QuasiNewton methods for solving nonsmooth equations: Generalized DennisMoré theorem and Broyden's update (with H.V. Ngai).
Journal of Convex Analysis, 25 (2018), No. 4, pp 10751104.
Abstract
Abstract:
In this paper, we study the quasiNewton method by using setvalued approximations for solving generalized equations without smoothness assumptions. The setvalued approximations appear naturally
when dealing with nonsmooth problems, or even in smooth cases, data in almost concrete applications are not exact. We present a generalization of the classical DennisMor\'{e} theorem, which gives a characterization
of the $q$superlinear convergence of the quasiNewton iterates. The local linear and superlinear convergences of the method, especially, a modification of the Broyden update method are investigated.
We present an example showing that the classical Broyden update method is no longer linearly convergent when the function involved in the nonlinear equation is not smooth. A modified version of the Broyden update
is proposed and its convergence is proved. These results are new, and can be considered as both an improvement and an extension of some results appeared recently in the literature on this subject.
Unbounded StateDependent Sweeping Processes with Perturbations in Uniformly Convex and qUniformly Smooth Banach Spaces,
(with B.K. Le).
Numerical Algebra, Control and Optimization (NACO AIMS), Volume 8, Number 1, pp. 8195 (2018). DOI: 10.3934/naco.2018005.
Abstract
Abstract:
In this paper, the existence of solutions for a class of first and second order unbounded statedependent
sweeping processes with perturbation in uniformly convex and $q$uniformly smooth Banach spaces are analyzed by using
a discretization method. The sweeping process is a particular differential inclusion with a normal cone to a moving set and is
of a great interest in many concrete applications.
The boundedness of the moving set, which plays a crucial role for the existence of solutions in many works in the literature,
is not necessary in the present paper. The compactness assumption on the moving set is also improved.
2017
An Extended FenchelLagrange Duality Approach and Optimality
Conditions for Strong Bilevel Programming Problems,
(with A. Aboussoror and F.E. Saissi).
SIAM Journal On Optimization (SIOPT), no. 2, 1230–1255 (2017).
Abstract
Abstract:
In this paper we give a conjugate duality approach for a strong
bilevel programming problem $(S)$. The approach is based on the
use of a regularization of problem $(S)$ and the socalled
FenchelLagrange duality. We first show that a regularized problem
of $(S)$ admits solutions and any accumulation point of a sequence
of regularized solutions solves $(S)$. Then, via this duality
approach, we provide necessary and sufficient optimality
conditions for the regularized problem. Finally, necessary and
sufficient optimality conditions are given for $(S)$. We note
that such an approach which allows us the application of the
FenchelLagrange duality to the class of strong bilevel
programming problems is new in the literature.
Stability of metric regularity with setvalued perturbations and application to
Newton's method for solving generalized equations, (with H. V. Ngai and N. Van Vu).
Journal SetValued and Variational Analysis, September 2017, Volume 25, Issue 3, pp 543–567.
Abstract
Abstract:
In this paper, we deal firstly with the question of the stability of the metric regularity under setvalued perturbation. By adopting the measure of closeness between two multifunctions,
we establish some stability results on the semilocal/local metric regularity. These results are applied to study the convergence of iterative schemes of Newtontype methods for solving generalized equations
in which the setvalued part is approximated. Some examples illustrating the applicability of the proposed method are discussed.
Nonconvex Sweeping Processes Involving Maximal Monotone Operators,
(with B.K. Le).
Optimization Optimization 66, no. 9, 1465–1486 (2017).
Abstract
Abstract:
By using a regularization method, we study in this paper the global existence and uniqueness property of a new variant of nonconvex
sweeping processes involving maximal monotone operators.
The system can be considered as a maximal monotone differential inclusion under a control term of normal cone type forcing the
trajectory to be always contained in the desired moving set. When the set is fixed, one can show that the unique solution is
rightdifferentiable everywhere and its rightderivative is rightcontinuous.
Nonsmooth Lyapunov pairs for this system are also analyzed.
Maximal Monotonicity and CyclicMonotonicity Arising in Nonsmooth Lur'e
Dynamical Systems,
(with A. Hantoute and B.K. Le).
Journal of Mathematical Analysis and Applications,
448 (2017), no. 1, 691–706
Abstract
Abstract:
We study a precomposition of a maximal monotone operator with linear mappings, which preserves the maximal monotonicity in the setting
of reflexive Banach spaces. Instead of using the adjoint of such linear operators, as in the usual precomposition, we consider
a more general situation involving operators which satisfy the socalled passivity condition. We also provide similar
analysis for the preservation of the maximal cyclic monotonicity. These results are applied to derive existence results for
nonsmooth Lur'e dynamical systems.
StrongWeak Nonlinear Bilevel Problems: Existence of Solutions in a Sequential Setting,
(with A. Aboussoror and F. Saissi).
Journal Setvalued and Variational Analysis, 25 (2017), no. 1, 113–132. DOI 10.1007/s1122801603694
Abstract
Abstract:
The paper deals with a strongweak nonlinear
bilevel problem $({S})$ which generalizes the wellknown weak and
strong ones. For such a problem, we first give a regularization based on the use of strict $\epsilon$solutions of the lower level problem.
Under appropriate assumptions, we prove the existence of solutions to the regularized problem $(S_{\epsilon})$. Then, we show that any
accumulation point of a sequence of regularized solutions solves the original problem $(S)$. The obtained result is an extension
of the one given in \cite{[3]} for weak bilevel programming problems. It also generalizes the result on the existence of solutions
given in \cite{[26]} for the linear finite dimensional case.
A duality approach and optimality conditions for simple convex bilevel programming problems, (with A. Aboussoror).
Pacific Journal of Optimization, Volume 13, Number 1, pp 123135 (2017).
Abstract
Abstract:
The paper deals with a convex bilevel programming problem
$({S})$ which never satisfies the Slater's condition. Using
$\epsilon$approximate solutions of the lower level problem, we
introduce a regularized bilevel problem $(S_{\epsilon})$ of $(S)$
that satisfies this condition. We show that $\inf
S_{\epsilon}\to\inf S$ when $\epsilon$ goes to zero and
that any
accumulation point of a sequence of regularized solutions solves
the original problem $(S)$. We provide optimality conditions for
the regularized problem via the FenchelLagrange duality. Then,
necessary optimality conditions are established for the solutions
of $(S)$ that are accumulation points of sequences of regularized
solutions. Finally, sufficient optimality conditions are
established for $(S)$.
Discontinuous Sweeping Process with ProxRegular Sets, (with F. Nacry and L. Thibault).
ESAIM: Control, Optimisation and
Calculus of Variations, Vol. 23, No 4 pp 12931329 (2017): https://doi.org/10.1051/cocv/2016053
Abstract
Abstract:
In this paper, we study the wellposedness (in the sense of existence and uniqueness of a solution) of a discontinuous
sweeping process involving proxregular sets in Hilbert spaces. The variation of the moving set is controlled by a
positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with
bounded variation is
achieved thanks to the Moreau's catchingup algorithm adapted to this kind of problem. Various properties and estimates of
jumps of the solution are also provided.
We give sufficient conditions to ensure the uniform proxregularity when the moving set is described by inequality constraints.
As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential
equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular
electrical circuits and control systems.
2016
Preservation of proxregularity of sets and application to constrained optimization, (with F. Nacry and L. Thibault).
SIAM Journal on Optimization SIOPT, Vol. 26, No 1, pp 448473 (2016)
Abstract
Abstract:
In this paper, we first provide counterexamples showing that sublevels of proxregular functions and
levels of differentiable mappings with Lipschitz derivatives may fail to be proxregular. Then, we prove the uniform
proxregularity of such sets under usual verifiable qualification conditions. The preservation of uniform proxregularity
of intersection and
inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given.
Implicit Euler TimeDiscretization of a Class of Lagrangian Systems with SetValued Robust Controller,
(with B. Brogliato and B.K. Le),
Journal of Convex Analysis 23 (2016), No. 1,
pp 2352. Abstract
Abstract:
A class of Lagrangian continuous dynamical systems with setvalued controller and subjected to a perturbation force
has been thoroughly studied in [S. Adly, B. Brogliato, B. K. Le, Wellposedness, robustness and stability analysis of a setvalued controller for Lagrangian systems, SIAM J. Control Optim.,
51(2), 15921614, 2013]. In this paper, we study the time discretization of these setvalued systems with an implicit Euler scheme.
Under some mild conditions, the wellposedness (existence and uniqueness of solutions) of the discretetime scheme, as well as the convergence of the sequences of discrete positions and velocities in finite steps are assured. Furthermore, the approximate piecewise linear function generated by these discrete sequences is shown to converge to the solution of the continuous time differential inclusion with order $\frac{1}{2}$.
Some numerical simulations on a twodegree of freedom example illustrate the theoretical developments.
Unbounded Second Order State Dependent Moreau's
Sweeping Processes in Hilbert Spaces, (with Ba Khiet Le).
Journal of Optimization Theory and Applications, (2016) 169 pp 407–423. DOI 10.1007/s1095701609052
Abstract
Abstract:
In this paper, an existence and uniqueness result of a second order sweeping process with
velocity in the moving set under perturbation in infinitedimensional Hilbert spaces
is studied by using an implicit discretization scheme. It is assumed that the moving set depends
on the time, the state and is allowed to be unbounded. The compactness assumption on the moving set is
improved comparing to previous works.
Our methodology is based on convex and variational analysis.
Newton's method for solving generalized equations: Kantorovich's and Smale's approaches,
(with H. V. Ngai and N. Van Vu).
Journal of Mathematical Analysis and Applications, 439 (2016), no. 1, 396–418
Abstract
Abstract:
In this paper, we study the Newtontype method for solving generalized equations involving setvalued maps in Banach spaces.
Kantorovichtype theorems (both local and global versions) are proved as well as the quadratic convergence of the Newton
sequence. We also extend both Smale's classical $(\alpha, \gamma)$theory to generalized equations.
These results are new and can be considered as an extension of many known ones in the literature for the classical nonlinear
case. Our approach is based on tools from variational analysis, where the metric regularity concept plays an important role in
our analysis.
Nonsmooth Lyapunov pairs for differential inclusions goverened by operators with nonempty interior domain, (with A. Hantoute and M. Théra)
Mathematical Programming Ser. B. 157, No 2, pp 349–374 (2016)
Abstract
Abstract:
The general theory of Lyapunov's stability of firstorder differential inclusions in Hilbert spaces has been studied by the authors in a previous work.
This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior.
This setting permits to have nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion.
It also allows some more explicit criteria for Lyapunov's pairs. Some consequences to the viability of closed sets are given, as well as some useful
cases relying on the continuity or/and convexity of the involved functions. Our analysis makes use of standard tools from convex and variational analysis.
Finitetime stabilization and H_\infty Control of nonlinear timevarying delay systems via output feedback,
(with Ta T.H. Trang and Vu N. Phat).
Journal of Industrial and Management Optimization, Volume 12, Issue 1,
2016 Pages 303315.
Abstract
Abstract:
This paper studies the robust finitetime $H_\infty$ control for a class of nonlinear systems with timevarying delay
and disturbances via ouput feedback. Based on the Lyapunov
functional method and a generalized Jensen integral inequality, novel delaydependent conditions for the existence of
output feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow
us to design the output feedback controllers which robustly stabilize the closedloop system in the finitetime sense.
An application to $H_\infty$ control of
uncertain linear systems with interval timevarying delay is also given. A numerical example is given to illustrate
the efficiency of the proposed method.
Optimal Guaranteed Cost Control of Nonlinear TimeVarying Delay Systems via Static Output Feedback, (with T.T. Trang and V.N. Phat).
Accepted for publication in Pacific Journal of Optimization (2016), Volume 12, Number 3, pp. 649667
Abstract
Abstract:
The optimal guaranteed cost control problem via static output feedback controller is ad dressed in this paper for a class of dynamical systems with
interval timevarying delays and nonlinear perturbations. By introducing a set of improved LyapunovKrasovskii functionals, a novel delaydependent
condition for output feedback guaranteed cost control with guar anteed exponential stability is derived in terms of linear matrix inequalities (LMIs).
Then, a design method of robust guaranteed cost control via output feedback controller is applied for uncertain linear systems. The design of output
feedback controllers can be carried out in a systematic and computationally efficient manner via the use of LMIbased algorithms. Numerical examples are
included to illustrate the effectiveness of the obtained result.
Nonsmooth Lur'e Dynamical Systems in Hilbert Spaces,
(with A. Hantoute and B.K. Le).
SetValued and Variational Analysis, March 2016, Volume 24, Issue 1, pp 1335.
First online: 04 July 2015
Abstract
Abstract:
In this paper, we study the wellposedness and stability analysis of setvalued Lur'e dynamical systems
in infinitedimensional Hilbert spaces. The existence and uniqueness results are established under the socalled passivity
condition. Our approach uses a regularization procedure for the term involving the maximal monotone operator.
The Lyapunov stability as well as the invariance properties are considered in detail.
In addition, we give some sufficient conditions ensuring the robust stability of the system in finitedimensional spaces.
The theoretical developments are illustrated by means of some examples dealing with nonregular electrical circuits.
This work extends and improves some of the recent results given in \cite{bg2,bg1,cs}.
Our methodology is based on tools from setvalued and variational analysis.
2015
Newton's method for solving inclusions using setvalued approximations, (with R. Cibulka and H.V. Ngai), SIAM Journal on Optimization, 25 (2015), no. 1, 159–184
Abstract
Abstract:
Results on stability of both local and global metric regularity under setvalued perturbations are presented.
As an application, we study (super)linear convergence of the Newtontype iterative process for solving generalized equations.
The possibility to choose setvalued approximations allows us to describe several iterative schemes in a unified way
(such as inexact Newton method, nonsmooth Newton method for semismooth functions, inexact proximal point algorithm, etc.).
Moreover, it also covers a forwardbackward splitting algorithm for finding a common zero of the sum of two multivalued (not necessarily monotone)
operators. Finally, a globalization of the Newton's method is discussed.
A new method for solving SecondOrder Cone Eigenvalue Complementarity Problem, (with H. Rammal) Journal of Optimization Theory and Applications, 165 (2015), no. 2, 563–585
Abstract
Abstract:
Eigenvalue complementarity problem EiCP with nonnegativity constraints has become a fruitful discipline within mathematical programming.
In this paper, we extend EiCP to problem where the nonnegative orthant, i.e., the Pareto cone is replaced by the product of second order cones SOC.
We reformulate such problem to find the roots of a semismooth function. Furthermore, we generalize the Lattice Projection Method LPM proposed first
in \cite{AR} to solve the second order cone eigenvalue complementarity problem SOCEiCP. The originality of this work, in comparison with \cite{AR},
is that we use a globalization of the semismooth Newton method SNM to approximate the Lorentz eigenvalues. Surprisngly, this kind of subject has never
been studied before due to the difficulty of this problem in the sense that the Lorentz spectrum is not always finite. Finally, LPM is then compared to
the semismooth Newton methods with line search: SNM$_{\min}$ and SNM$_{\rm FB}$, by using the performance profiles \cite{Cops, perf} as a comparison tool.
The numerical experiments highlight that the LPM solver is efficient and robust for solving SOCEiCP.
2014
On onesided Lipschitz stability of setvalued contractions,
(with A.L. Dontchev and M. Théra), Numerical Functional Analysis and Optimization,
35 (2014), no. 79, 837–850
Abstract
Abstract:
We show that a result by T. C. Lim [On fixedpoint stability for setvalued contractive mappings with applications to generalized differential
equations, J. Math. Anal. Appl. 110 (1985) 436441] can be sharpened significanly
by using a generalization of a theorem by Arutyunov regarding fixed points of composition of mappings.
A global version of the LyusternikGraves theorem is a corollary of this estimate as well. We apply the generalization of Lim's result
to derive onesided Lipschitz properties
of the solution mapping of a differential inclusion with a parameter.
Quantitative stability of a generalized equation. Application to nonregular electrical circuits,
(with R. Cibulka)
Journal of Optimization Theory and Applications JOTA, 160 (2014), no. 1, 90–110
Abstract
Abstract:
The paper is devoted to the study of several stability properties (such as Aubin property, calmness and isolated calmness) of a special nonmonotone
generalized equation. The theoretical results are applied in the theory of nonregular electrical circuits involving electronic devices like ideal diode, practical
diode, and DIACs (DIode Alternating Current).
Convex Sweeping Process in the framework of Measure Differential Inclusions and Evolution Variational Inequalities,
(with T. Haddad and L. Thibault), Mathematical Programming,
148 (2014), no. 12, Ser. B, pages: 5–47
Abstract
Abstract:
In this paper, we analyze and discuss the wellposedness of two new variants of the socalled sweeping process, introduced by J.J. Moreau in the early
70's \cite{More71} with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving
convex subset $C(t)$, supposed to have a bounded retraction, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed
differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for a which a large analysis is made,
concerns a first order sweeping process with velocity in the moving set $C(t)$. This class of problems subsumes as a particular case, the evolution
variational inequalities (widely used in applied mathematics and unilateral mechanics \cite{DL}). Assuming that the moving subset $C(t)$ has a continuous
variation for every $t\in [0,T]$ with $C(0)$ bounded, we show that the problem has at least a Lipschitz continuous solution. The wellposedness of this class
of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process for the
study of vector hysteresis operators in the elastoplastic model \cite{Krej91}, the planning procedure in mathematical economy \cite{Henr} and to nonregular
electrical circuits containing nonsmooth electronic devices like diodes \cite{abb}. The theoretical results are supported by some numerical simulations to
prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like the other papers in
this collection, we show in this presentation how elegant modern convex analysis was influenced by the Moreau's seminal work.
Stability and invariance results for a class of nonmonotone setvalued Lur’e dynamical systems, (with B. K. Le)
Applicable Analysis, 93 (2014), no. 5, 1087–1105. .
Abstract
Abstract:
In this paper, we analyze the wellposedness, stability and invariance results for a class of nonmonotone setvalued Lur'e dynamical system which has
been widely studied in control and applied mathematics (see \cite{l}). Many recent researches deal with the case when the setvalued part is the
subdifferential of some proper, convex, lower semicontinuous function in order to use the nice properties of maximally monotone operators.
But in practice, particularly in electronics, there are some devices such as diac, silicon controller rectifier (SCR)...of which the voltagecurrent
characteristics are not monotone but only locally hypomonotone. This fact motivates us to write the paper which is organized as follows: firstly the
existence and uniqueness of solutions are proved by using Filippov's method and local hypomonotonicity; then the stability analysis and generalized
LaSalle's invariance principle are presented. The theoretical results are supported by numerical simulations for some examples in electronics.
Our methology is based on nonsmooth and variational analysis.
Generalized SemiInfinite Programming: Optimality Conditions Involving Reverse Convex Problems,
(with A. Aboussoror).
Numerical Functional Analysis and Optimization, 35 (2014), no. 79, 816–836
Abstract
Abstract:
The paper deals with a generalized
semiinfinite programming problem $(S)$. Under appropriate
assumptions, for such a problem we give necessary and sufficient
optimality conditions via reverse convex problems. In particular,
a necessary and sufficient optimality condition reduces the
problem $(S)$ to a $\min$$\max$ problem constrained with compact
convex linked constraints.
2013
 Wellposedness, Robustness and Stability Analysis of a SetValued Controller for Lagrangian Systems, (with B. Brogliato and B. K. Le),
SIAM Journal on Control and Optimization 51 (2013), no. 2, 1592–1614
Abstract
Abstract:
This paper deals with the analysis of a class of nonsmooth robust controllers for lagrangian systems with nontrivial mass matrix.
First the existence and uniqueness of solutions are analyzed, then the Lyapunov stability, the KrasovskiiLaSalle invariance principle and
finitetime convergence properties are studied.
 A New Method for Solving Pareto Eigenvalue Complementarity Problems, (with H. Rammal),
Computational Optimization and Applications 55, No 3, pp 703731 (2013).
Abstract
Abstract:
In this paper, we introduce a new method, called the Lattice Projection Method (LPM), for solving eigenvalue complementarity problems.
The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues
and eigenvectors of the complementarity problems. The LPM is compared to SNM$_{\rm min}$ and SNM$_{\rm FB}$, two methods widely discussed in the literature
for solving nonlinear complementarity problems, by using the performance profiles as a comparing tool. The performance measures, used to analyze
the three solvers on a set of matrices mostly taken from the Matrix Market, are computing time, number of iterations, number of failures and maximum
number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving
eigenvalue complementarity problems. Finally, Pareto bieigenvalue complementarity problems were solved numerically as an application to confirm the
efficiency of our method.
 Variational Analysis and Generalized Equations in Electronics. Stability and Simulation Issues, (with R. Cibulka and H. Massias),
SetValued and Variational Analysis 21 (2013), no. 2, 333–358.
Abstract
Abstract:
The paper is devoted to the study of the Aubin/Lipschitzlike property and the isolated calmness of a particular nonmonotone generalized equation arising in electronics.
The variational and nonsmooth analysis is applied in the theory of nonregular electrical circuits involving electronic devices like ideal diodes, practical diodes,
DIACs, silicon controlled rectifiers (SCR), and transistors.
We also discuss the relationship of our results to the ones using classical techniques from (smooth) analysis and provide a simulation for
several simple electrical circuits which are chosen in order to cover the most common nonsmooth elements in electronics.
The simulations of the electrical circuits discussed in this paper are performed by using Xcos (a component of Scilab).
 On some dynamic thermal non clamped contact problems, (with O. Chau), Mathematical Programming
serie B, No 12, pp 526 (2013)
Abstract
Abstract:
We study a class of dynamic thermal subdifferential contact problems with friction, for long memory viscoelastic materials,
without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality,
coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general
results on first order evolution inequality, with monotone operators and fixed point methods. Finally a fully discrete scheme for numerical approximations is provided, and corresponding various numerical computations in dimension two will be given.
 Qualitative Stability of a Class of NonMonotone Variational Inclusions. Application in Electronics, (with J.V. Outrata),
Journal of Convex Analysis 20, 1 (2013)
Abstract
Abstract:
The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated calmness) of a special
nonmonotone variational inclusion. We provide a characterization of these properties in terms of the problem data and show their importance
for the design of electrical circuits involving nonsmooth and nonmonotone electronic devices like DIAC (DIode Alternating Current).
Circuits with other devices like SCR (Silicon Controlled Rectifiers), Zener diodes, thyristors, varactors and transistors can be analyzed in the same way.
2012
 Stability Analysis and Attractivity Results of a DCDC Buck Converter, (with D. Goeleven and B. K. Le),
Set Valued and Variational
Analysis (2012) 20:331353
Abstract
Abstract:
Using tools from setvalued and variational analysis, we propose a mathematical formulation for a power DCDC 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 implicit schemes.
 Nonsmooth Lyapunov pairs for infinitedimensional firstorder differential inclusions, (with A.Hantoute and M. Thera),
Nonlinear
Analysis: Theory, Methods and Applications 75, 3 (2012) 9851008
Abstract
Abstract:
The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated
to firstorder differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator.
The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described
in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature.
Our approach is based on advanced tools of variational analysis and generalized differentiation.
 Solvability of a class of thermal dynamical contact problems with subdifferential conditions, (with O. Chau and M. Rochdi),
Numerical Algebra, Control and Optimization 2, 1, pp 91104 (2012).
Abstract
Abstract:
We study a class of dynamic thermal subdifferential contact problems with friction, for long memory viscoelastic materials, which can be put into a general model of
system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first
order evolution inequality, with monotone operators and fixed point methods.
2011
 A FenchelLagrange Duality Approach for a Bilevel Programming Problem with ExtremalValue Function, (with A. Aboussoror),
Journal of Optimization Theory and Applications 149, 2 (2011) 254268
Abstract
Abstract:
In this paper, for a bilevel programming problem (S) with an extremalvalue function, we first give its FenchelLagrange dual problem. Under appropriate assumptions, we show that a strong duality holds between them. Then, we
provide optimality conditions for (S) and its dual. Finally, we show that the resolution of the dual problem is equivalent to the resolution of a onelevel convex minimization problem.
 Finitetime Lyapunov stability analysis of evolution variational inequalities, (with K. Addi and H. Saoud),
Discrete and Continuous Dynamical Systems  Series A 31, 4 (2011) 10231038
Abstract
Abstract:
Using Lyapunov's stability and LaSalle's invariance principle for nonsmooth dynamical systems, we establish some conditions for finitetime stability of evolution variational inequalities. The theoretical results are illustrated by some examples drawn from electrical circuits involving nonsmooth elements like diodes.
 A nonsmooth algorithm for coneconstrained eigenvalue problems, (with A. Seeger), Computational
Optimization and Applications 49, 2 (2011) 299318
Abstract
Abstract:
We study several variants of a nonsmooth Newtontype algorithm for solving an eigenvalue problem of the form Kx(Ax−Bx)K+ Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics.
The symbol K refers to a closed convex cone in the Euclidean space R^n and (A,B) is a pair of possibly asymmetric matrices of order n. Special attention is paid to the case in which K is the nonnegative orthant of R^n .
The more general case of a possibly unpointed polyhedral convex cone is also discussed in detail.
 Weak Nonlinear Bilevel Problems : Existence of Solutions via Reverse Convex and Convex Maximization Problems,
(with A. Aboussoror and V. Jalby), Journal of Industrial and Management
Optimization 7, 3 (2011) 559571
Abstract
Abstract:
In this paper, for a class of weak bilevel programming problems we provide sufficient conditions guaranteeing the existence of global solutions. These conditions are based on the use of reverse
convex and convex maximization problems.
2010
 Optimal control of a quasivariational obstacle problem, (with M. AitMansour and M. Bergounioux),
Journal of Golobal Optimization,
Vol. 47, Number 3, pp 421435 (2010).
Abstract
Abstract:
We consider an optimal control where the statecontrol relation is given by a quasivariational inequality, namely a generalized obstacle problem. We give an existence result for solutions to such a problem.
The main tool is a stability result, based on the Moscoconvergence theory, that gives the weak closeness of the controltostate operator. We end the paper with some examples.
2009
 Periodic solutions of Evolution Variational Inequalities: a method of guiding functions,
(with D. Goeleven and M. Théra), Chinese Annals of Mathematics,
Series B , no 3, pp 261272 (2009).
2008
 A sensitivity analysis of a class of semicoercive variational inequalities using recession tools, (with K. Addi, D. Goeleven and H. Saoud),
Journal of Global Optimization, vol. 40, no. 1, pp. 727 (2008).
2007
 A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of nonregular circuits in electronics,
(with K. Addi, B. Brogliato and D. Goeleven), Nonlinear Analysis: Hybrid Systems and Applications, N° 1,
pp 315324 (2007).
 A continuation method for a class of periodic evolution variational inequalities, (with D. Goeleven and M. Théra),
Some Topics in Industrial and Applied Mathematics. Series in Contemporary Applied Mathematics CAM 8, pp 128 (2007).
 Stability analysis of filtered massspring systems, (with A. Ahmad, D. Ghazanfarpour and O. Terraz),
Proceedings of theory and practice of computer graphics , 2007, pp 4552 (Bangor, England).
 Existence Results for a Class of Periodic Evolution Variational Inequalities, (with D. Goeleven and M. Théra),
Chinese Annals of Mathematics  Series B, Volume 28, Number 6, pp 629650 (2007).
2006
 Finite time stabilization of nonlinear oscillators subject to dry friction, (with H. Attouch, A. Cabot)
Progresses in Nonsmooth Mechanics and Analysis (edited by P. Alart, O. Maisonneuve and R.T. Rockafellar), Advances in Mathematics and Mechanics, Kluwer, pp 289304, (2006).
 Attractivity Theory for Second Order Nonsmooth Dynamical Systems with application to dry friction,
Journal of Mathematical Analysis and Applications , 322 pp 10551070 (2006).
Abstract
Abstract:
In this paper, we study the attractivity properties of the set of stationary solutions for a general class of second order nonsmooth
dynamical system involving friction term. Sufficient conditions for the local attractivity of the set of stationary solutions are given in the cas of dry friction and negative viscous damping. An estimation of the attraction domain is also given in this case. Applications can be found in unilateral mechanics.
 Sensitivity analysis of solutions to a class of quasivariational inequalities, (with M. AitMansour and L. Scrimali),
Bulletino Della Unione Mathematica Italiana (BUMI) pp 767772 (2006).
 Stability in frictional unilateral elasticity revisited : an application of the theory of semicoercive variational inequalities,
(with E. Ernst et M. Théra),
Proceeding of American Institute of Physics, Vol. 835, pp 111 (2006).
 The approach of Moreau and Panagiotopoulos: use it in electronics, (with K. Addi, B. Brogliato and D. Goeleven),
accepté dans Proceeding of the International Conference on
Nonsmooth/Nonconvex Mechanics with Applications in Engineering (NNMAE2006) pp 471478 (2006).
 Stability of linear semicoercive variational inequalities in Hilbert spaces: application to
the SignoriniFichera Problem, Journal of Nonlinear and Convex Analysis, N° 3, pp 325334 (2006).
Abstract
Abstract:
In this paper we show how recent results concerning the stability of semicoercive variational inequalities on reflexive Banach spaces, obtained in 25 can be applied to establish the stability of the semicoercive SignoriniFichera problem with respect to small perturbations.
 A nonsymmetric linear complementarity problem to solve a quasistatic rolling frictional contact problem,
(with K. Addi, D. Goeleven and M. Théra), Journal of Nonlinear and Convex Analysis , N°3, pp 315324 (2006).
Abstract
Abstract:
A simple approach and an algorithm are proposed to solve the quasistatic rolling frictional contact problem between an elastic cylinder and a flat 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 complementarity problem which is solved using Lemke's algorithm.
 Stabilisation par filtrage de méthodes d'intégration explicite, (with A. Ahmad, D. Ghazanfarpour and O. Terraz),
Journées AFIG (Association Française d'Informatique Graphique) pp 7380 (2006).
2004
 Norm Closure of the Barrier cone in normed linear spaces, (with E. Ernst and M. Théra),
Proceeding of the American Mathematical Society , 132 no 10, pp 29112915 (2004).
Abstract
Abstract:
The aim of this note is to characterize the normclosure of the barrier cone of a closed convex set in an arbitrary normed linear space by means
of a new geometric object, the temperate cone.
 Wellpositioned closed convex sets and wellpositioned closed convex functions, (with E. Ernst and M. Théra),
Journal of Global Optimization , 29 (4), pp 337351 (2004).
 A stability Theory for secondorder nonsmooth dynamical systems with application to friction problems, (with D. Goeleven),
Journal de Mathématiques Pures et Appliquées , Vol. 83, pp 1751 (2004).
Abstract
Abstract:
A LaSalle's Invariance Theory for a class of firstorder evolution variational inequalities is developed. Using this approach,
stability and asymptotic properties of important classes of secondorder dynamic systems are studied. The theoretical results of
the paper are supported by examples in nonsmooth Mechanics and some numerical simulations.
2003
 On the closedness of the algebraic difference of closed convex sets, (with E. Ernst and M. Théra),
Journal de Mathématiques Pures et Appliquées, Vol. 82, 9, pp 12191249 (2003).
Abstract
Abstract:
We characterize in a reflexive Banach space all the closed convex sets C_{1} containing no lines for which the condition C_{1}^{∞} ∩ C_{2}^{∞} = ( 0 ); ensures the closedness of the algebraic difference C_{1}  C_{2} of all closed convex sets C_{2}.
We also answer a closely related problem: determine all the pairs C_{1} ,C_{2} of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbations of C_{1} and C_{2} remains closed. As an application, we state the broadest setting for the strict separation theorem in a reflexive Banach space.
2002
 Stability of Noncoercive Variational Inequalities, (with E. Ernst and M. Théra),
Communication in Contemporary Mathematics ,4 No 1, pp 145160 (2002).
 On the converse of the Dieudonné theorem in reflexive Banach spaces, (with E. Ernst and M. Théra),
Cybernetics System Anal., Special Issue in memorium Prof. N.B. Pshenitchnyi, 38 No 3, pp 3439 (2002).
2001
 A Characterization of Convex and Semicoercive Functionals, (with E. Ernst and M. Théra)
Journal of Convex Analysis , Volume 8, No 1, pp 127148 (2001).
Abstract
Abstract:
In this paper we prove that, if every small L^{∞}perturbation of a proper, convex, l.s.c., extendedrealvalued functional defined on a reflexive Banach space attains its minimum value, then this functional is semicoercive.
 Satbilité de l'ensemble des solutions d'une inéquation variationnelle non coercive, (with E. Ernst and M. Théra),
Compte Rendu de l'Académie des Sciences (CRAS), T. 333 Série I, pp 409414 (2001).
2000
 A Discretization Theory for a Class Semicoercive Unilateral Problems, (with D. Goeleven),
Numerishe Mathematik
, 87, pp 134 (2000).
Abstract
Abstract:
In this paper, we present a convergence analysis applicable to the approximation of a large class of semicoercive variational inequalities.
The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finiteelement approximations of semicoercive unilateral
problems in mechanics are discussed. In particular, a SignoriniFichera unilateral contact model and some obstacle problem with frictions are studied.
The theoretical conditions proved are in good agreement with the numerical ones.
 Periodic solutions for second order differential equations involving nonconvex superpotentials, (with D. Motreanu)
Journal of Global Optimization, 17, pp 917 (2000).
Abstract
Abstract:
The paper establishes the existence of a nonconstant periodic solution of a general second order nonautonomous
Hamiltonian system with discontinuous nonlinearities. The multiplicity of solutions is also studied.
 Location of eigensolutions to variationalhemivariational inequalities, (avec D. Motreanu),
Journal of Nonlinear and Convex analysis , Vol. 1, No 3, pp 255270, (2000).
1999
 Solvability of generalized nonlinear symmetric variational inequalities, ( with W. Oettli),
Bullettin of the Austrelian Mathematical Society, Ser. B (40), pp 289300 (1999).
Abstract
Abstract:
This paper deals with the study of a general class of nonlinear variational inequalities. An existence result is given, and a perturbed iterative scheme is analyzed for solving such problems.
1998
 Critical points for nonsmooth energy functions and applications, (with G. Buttazzo and M. Théra),
Nonlinear Analysis, Theory Methods and Applications , Vol. 32, No 6, pp. 711718 (1998).
 Unilateral Hamiltonian Systems: a survey on the inequality approach, (with D. Goeleven and D. Motreanu),
Advances in Nonlinear Variational Inequalities , 1 , no. 1, pp 1126 (1998)
Abstract
Abstract:
The aim of this paper is to provide a survey of the results related to the mathematical study of a new class of unilateral problems involving nonconvex and nonsmooth energy functions. More precisely, we are concerned with those problems which have as variational formulations the socalled hemivariational inequalities. Until 1981 most of the unilateral problems studied in Mechanics were expressed in term of variational inequalities which are expressions of the principle of virtual work or power in inequality form. That means that only material laws and boundary conditions formulated through convex superpotentials, i.e. subdifferentials of convex functionals could be considered. One knows also that the convexity of an energy function entails the monotonocity of the underlying stressstrain or forcedisplacement (velocity) relations. In order to overcome these restricting conditions of monotonicity in the mathematical formulation of either boundary conditions or material laws, P.D. Panagiotopoulos has introduced the notion of nonconvex superpotential by using the Clarke subgradient for locally Lipschitz functions. The resulting variational expressions of the principle of virtual work or power are inequalities called hemivariational inequalities. The hemivariational inequality approach has now been proved to be very efficient to describe various problems in Unilateral Mechanics such as for instance: laminated plates problems, nonmonotone semipermeable problems, adhesive grasping problems in robotics, loadingunloading problems, etc.
1997
 Periodic and homoclinic solutions for a class of unilateral problems, (with D. Goeleven and D. Motreanu),
Discrete and Continuous Dynamical Systems, vol. 3, Number 4, October (1997).
Abstract
Abstract:
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 nonsmooth energy functional.
1996
 Recession mappings and noncoercive variational inequalities, (with D. Goeleven and M. Théra),
Journal of Nonlinear Analysis, Theory Methods and Applications , Vol. 26, No. 9, pp 15731603 (1996).
Abstract
Abstract:
In this paper, we present a quite general existence theory
applicable to several class of noncoercive variational inequalities and
related variational problems. This result is based on the
asymptotic behavior of the sets and the operators which are
involved in the variational inequality. Applications are then
given to several strongly nonlinear boundaries problems at
resonance, to the NavierStockes equations, to an obstacle
problem, to the unilateral buckling of a simply supported
beam and to the unilateral buckling of a thin elastic plate.
Keywords: Maximal monotone operator, subdifferential operator, recession function, noncoercive variational inequality, NavierStokes equations, puffedup membrane, elastic plate,unilateral buckling.
 A perturbed iterative method for a general class of variational inequalities, Serdica Mathematical Journal
, Vol. 22, pp 6982 (1996).
Abstract
Abstract:
The generalized WienerHopf equation and the approximation methods are
used to propose a perturbed iterative method to compute the solutions of
a general class of nonlinear variational inequalities.
 Iterative algorithms and sensivity analysis for a general class of variational inclusions,
Journal of Mathematical Analysis and Applications , 201, pp 609630 (1996).
Abstract
Abstract:
In the present paper, we study a perturbed iterative method for solving
a general class of variational inclusions. An existence result which
generalizes some known results in this field, a convergence result and
a new iterative method are given. We also prove the continuity of the
perturbed solution to a parametric variational inclusion problem.
Several special cases are discussed.
1995
[J3] 
S. Adly and D. Goeleven.
Periodic solutions for a class of hemivariational inequalities,
Communications on Applied Nonlinear Analysis ,
2, No. 2, pp 4758 (1995).
Abstract
Abstract:
The field of inequality problems has seen a considerable
development in mathematics and unilateral mechanics.
Particularly, the theory of variational inequalities is now a
welldeveloped theory in mathematics.
The mechanical meaning of a variational inequality is given by
the formulation of the principle of virtual work when a
monotone stressstrain or reactiondisplacement condition hold.
In the case of lack of monotonicity of these underlying
stressstrain or reactiondisplacement conditions, the
variational expressions of the principle of virtual work are
no longer variational inequalities. Another type of inequality
expression arises as variational formulation of the problem.
These inequalities are called hemivariational inequalities and
were born only 12 years ago. The idea of these new inequality
methods are due to Panagiotopoulos. For more details concerning
these approach, we refer the reader to the book of P.D.
Panagiotopoulos and of Naniewicz & Panagiotopoulos.
It is clear that much could be done to extend the existence
theory of variational inequality to this case.
This paper starts with the study of a system of differential
equations involving discontinuous nonlinearities. The method
employed is based on nonsmooth critical point theory of
Chang.
We shall show that our approach contributes to the mathematical
theory of hemivariational inequalities.

[J2] 
S. Adly and D. Goeleven.
Homoclinic orbits for a class of hemivariational inequalities,
Journal of Applicable Analysis ,
Vol. 58, pp 229240 (1995).
Abstract
Abstract:
The hemivariational inequalities approach has been now proved to be
very efficient to describe the behavior of several complex structures
as multilayered plates, adhesive joints, composite structures, etc.
For details concerning the physical problems, we refer the reader to
the works of P. D. Panagiotopoulos. As a consequence of the contribution
s of P. D. Panagiotopoulos, the study of hemivariational inequalities
has emerged as an interesting branch of applied mathematics and this topi
c is now the object of the attention of several engineers and
mathematicians as P. D. Panagiotopoulos, Z. Naniewicz, D. Goeleven &
M. Théra, S. Adly & D. Goeleven.
Using a new compact imbedding theorem of C. De Coster and M. Willem,
we prove the existence of homoclinic orbits for a class of
hemivariational inequalities.

[J1] 
S. Adly, D. Goeleven and M. Théra.
Recession methods in monotone variational hemivariational inequalities,
Journal of Topological Methods in Nonlinear Analysis ,
Vol. 5, No. 2, pp. 397409, (1995).
Abstract
Abstract:
In this note, using the well known Ky Fan minimax principle,
we prove an existence theorem applicable to a large class of
variational hemivariational inequalities involving
hemicontinuous monotone operators.
