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Books

A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics.
SpringerBriefs in Mathematics.
Available on February 2018.
About this book
About this book:
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
Refereed Publications
2018
Proxregularity approach to generalized equations and image projection,
(with F. Nacry and L. Thibault).
Accepted for publication in ESAIM: Control, Optimisation and Calculus of Variations. 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.
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 130 (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.
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.
Unbounded StateDependent Sweeping Processes with Perturbations in Uniformly Convex and qUniformly Smooth Banach Spaces,
(with B.K. Le).
Accepted for publication in Numerical Algebra, Control and Optimization (NACO AIMS).
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.
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
 Periodic solutions for a class of hemivariational inequalities, (with D. Goeleven),
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.
 Homoclinic orbits for a class of hemivariational inequalities, (with D. Goeleven),
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.
 Recession methods in monotone variational hemivariational inequalities, (with D. Goeleven and M. Théra)
,
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.