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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, second-order 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

• Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping, (with Hedy Attouch). Submitted on December 2019) . Abstract

  Abstract: In a Hilbert space $H$, we introduce a new class of proximal-gradient 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, first-order 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 Lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of a variant of the Nesterov accelerated method with dry friction and Hessian-driven damping.

• Sensitivity analysis of maximal monotone inclusions via the proto-differentiability of the resolvent operator, (with R.T. Rockafellar). Submitted on November 2019) . 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 proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative 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 semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual 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 Kuhn-Tucker set.

• Finite convergence of proximal-gradient inertial algorithms with dry friction damping, (with Hedy Attouch). Submitted on November 2019) . Abstract

  Abstract: In a Hilbert space $H$, based on inertial dynamics with dry friction damping, we introduce a new class of proximal-gradient algorithms with finite convergence properties. The function $f:H \to \R$ to minimize 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. This algorithm naturally occurs as a discrete temporal version of an inertial differential inclusion involving viscous and dry friction together. The convergence results tolerate the presence of errors, under the sole assumption of their asymptotic convergence to zero. Then, 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 new splitting methods. Numerical experiments are given for Lasso-type problems. The performance profiles, as a comparison tool, highlight the effectiveness of two variants of the Nesterov accelerated method with dry friction.

• New metric properties for prox-regular sets, (with F. Nacry and L. Thibault). Submitted on November 2019) . Abstract

  Abstract: In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel 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 prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular 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 prox-regular sets have good properties known in convex analysis.

• Douglas-Rachford splitting algorithm for solving state-dependent maximal monotone inclusions, (with Le Ba Khiet). Submitted on November 2019) . Abstract

  Abstract: In this paper, we provide a new application of Douglas-Rachford 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.

• The derivative of a parameterized mechanical contact problem with a Tresca's friction law involves Signorini unilateral conditions, (with Loïc Bourdin and Fabien Caubet). Submitted on November 2019) . Abstract

  Abstract: The present paper investigates the sensitivity analysis, with respect to right-hand 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 {\it twice epi-differentiability}, 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 epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.

• Dennis-Moré condition for set-valued vector fields and the superlinear convergence of Broyden updates in Riemannian manifolds, (with Huynh Van Ngai and Van Vu Nguen) Submitted on April 2019 . Abstract

  Abstract: This paper deals with the quasi-Newton type scheme for solving generalized equations involving point-to-set 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 Dennis-Moré theorem in [1] as well as the Riemannian Dennis-Moré condition established in the work [2]. Furthermore, we also apply these results to consider the convergence of a Broyden-type update for the problem of solving generalized equations in Riemannian context. Our results are new even for classical equations defined by single-valued vector fields.
  
  [1] J. E. Dennis and J.J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Mathematics of Computation, 28 (1974), pp. 549--560.
  [2] K.A. Gallivan, C.Qi, and P.-A. Absil}, High-Performance Scientific Computing: Algorithms and Applications, Springer London, 2012, ch. A Riemannian Dennis-Moré Condition, pp.281--293.
  

• Friction models in the framework of set-valued and convex analysis, (with Daniel Goeleven and Rachid Oujja) Submitted on January 2019 . Abstract

  Abstract: Most models of friction are nonsmooth in the sense that the functions involved in the models are discontinuous. The pioneering works of Jean-Jacques Moreau (1923-2014) and Panagiotis D. Panagiotopoulos (1950-1998) catalyzed the development of a mathematical framework applicable to the study of nonsmooth mechanical problems in using advanced results of modern convex analysis and set-valued analysis. The approach of Moreau and Panagiotopoulos can in particular be used to write a precise and rigorous mathematical model describing the friction force and the stick-slip phenomena. This approach using set-valued functions left aside the complicated transition processes between "stick" and "slip" but lead to rigorous mathematical models like differential inclusions and variational inequalities. In this expository paper, we summarize the approaches dealing with friction trough different models.
  

Refereed Publications

2020

• Well-Posedness of nonconvex degenerate sweeping process via unconstrained evolution problems, (with T. Haddad) Nonlinear Analysis: Hybrid Systems, 36 (2020) https://doi.org/10.1016/j.nahs.2019.100832 . Abstract

  Abstract: The main concern of this paper is the study of degenerate sweeping process involving uniform prox-regular sets via an unconstrained differential inclusion by showing that the sets of solutions of the two problems coincide. This principle of reduction to unconstrained evolution problem can be seen as a penalization of the subdifferential of the distance function. Using this reduction technique, an existence and uniqueness result of a Lipschitz perturbed version of the degenerate sweeping process is proved in the finite dimensional setting. An application is given to quasistatic unilateral dynamics in nonsmooth mechanics where the moving set is described by a finite number of inequalities. We provide sufficient verifiable conditions ensuring both the prox-regularity and the Lipschitz continuity with respect to the Hausdorff distance of the moving set.

• A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction, (with Daniel Goeleven) Accepted for publication in Evolution Equations and Control Theory (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 well-posed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results.
  

2019

• A coderivative approach to the robust stability of composite parametric variational systems. Applications in nonsmooth mechanics. Journal of Optimization Theory and Applications, 180 (2019), no. 1, pp 62-90.
  https://doi.org/10.1007/s10957-018-1293-6 . Abstract

  Abstract: The main concern of this paper is to investigate the Lipschitzian-like stability property (namely Aubin property) of the solution map of possibly nonmonotone variational systems with composite superpotentials. Using Mordukhovich coderivative criterion and a second-order 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). Accepted for publication in Set-Valued and Variational Analysis.
  https://doi.org/10.1007/s11228-019-00513-4 . 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 semi-infinite programming problems via a convex decomposition, (with A. Aboussoror and S. Salim) Accepted for publication in Optimization . Abstract

  Abstract: We present an extended conjugate duality for a generalized semi-infinite 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 semi-infinite programming.

• On a decomposition formula for the resolvent operator of the sum of two set-valued maps with monotonicity assumptions, (with L. Bourdin) Applied Mathematics and Optimization https://doi.org/10.1007/s00245-019-09599-6 . 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 set-valued 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 Douglas-Rachford 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) Accepted for publication in the Journal Pure and Applied Functional Analysis . 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 Fenchel-Lagrange duality approach and optimality conditions for strong bilevel programming problems, SIAM J. Optim. Vol. 27, No. 2, pp. 1230-1255, 2017.

• Well-posedness of discontinuous second-order nonconvex state-dependent sweeping processes, (with F. Nacry). Applied Mathematics and Optimization, 79 (2019), no. 2, 515-546. https://doi.org/10.1007/s00245-017-9446-9 . Abstract

  Abstract: In this paper, we study the existence of solutions for a time and state-dependent discontinuous nonconvex second order sweeping process with a multivalued perturbation. The moving set is assumed to be prox-regular, relatively ball-compact with a bounded variation. The perturbation of the normal cone is a scalarly upper semicontinuous convex valued multimapping satisfying a linear growth condition possibly time-dependent. As an application of the theoretical results, we investigate the theory of evolution quasi-variational inequalities.

• Time-dependent 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/s00033-019-1084-4 . Abstract

  Abstract: We consider a class of time-dependent 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 time-dependent 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). Set-valued and Variational Analysis, 27 (2019), no. 2, pp 549-568.
  http://dx.doi.org/10.1007/s11228-019-0505-z . 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 semi-convex functions, we give some continuity properties of the solution set associated to these problems. The particular case of the parametric sweeping process involving prox-regular 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.

• On Evolution quasi-Variational Inequalities and Implicit state-dependent Sweeping Processes, (with T. Haddad) Accepted for publication in Discrete and Continuous Dynamical Systems-S . Abstract

  Abstract: In this paper, we study a variant of the state-dependent 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 quasi-variational 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 quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lispchitz constant of the moving set, with respect to the state, is required.

• On a decomposition formula for the proximal operator of the sum of two convex functions, (with L. Bourdin and F. Caubet). Journal of Convex Analysis 26 (2019), No. 3 . Abstract

  Abstract: The main result of the present theoretical paper is an original decomposition formula for the proximal operator of the sum of two proper, lower semicontinuous and convex functions~$f$ and~$g$. For this purpose, we introduce a new operator, called {\it $f$-proximal operator of~$g$} and denoted by $\prox^f_g$, that generalizes the classical notion. Then we prove the decomposition formula~$\prox_{f+g} = \prox_f \circ \prox^f_g$. After collecting several properties and characterizations of~$\prox^f_g$, we prove that it coincides with the fixed points of a generalized version of the classical Douglas-Rachford operator. This relationship is used for the construction of a weakly convergent algorithm that computes numerically this new operator~$\prox^f_g$, and thus, from the decomposition formula, allows to compute numerically~$\prox_{f+g}$. It turns out that this algorithm was already considered and implemented in previous works, showing that~$\prox^f_g$ is already present (in a hidden form) and useful for numerical purposes in the existing literature. However, to the best of our knowledge, it has never been explicitly expressed in a closed formula and neither been deeply studied from a theoretical point of view. The present paper contributes to fill this gap in the literature. Finally we give an illustration of the usefulness of the decomposition formula in the context of sensitivity analysis of linear variational inequalities of second kind in a Hilbert space.

• State-dependent implicit sweeping process in the framework of quasistatic evolution quasi-variational inequalities, (with T. Haddad and B.K. Le). Journal of Optimization Theory and Applications, https://doi.org/10.1007/s10957-018-1427-x . Abstract

  Abstract: This paper deals with the existence and uniqueness of solutions for a class of state-dependent 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 hypomonotonicity-like 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 state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

• Lyapunov Stability of Differential Inclusions Involving Prox-Regular Sets via Maximal Monotone Operators, (with A. Hantoute and Bao Tran Nguyen). Journal of Optimization Theory and Applications, https://doi.org/10.1007/s10957-018-1446-7 . 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 prox-regular 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 Luenberger-like 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 prox-regular 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 prox-regular sets.

2018

• Sensitivity analysis of variational inequalities via twice epi-differentiability and proto-differentiability of the proximity operator, (with L. Bourdin). SIAM Journal on Optimization (SIOPT), 28 (2018), no. 2, 1699-1725. . 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 second-order differentiations of the involved operators and functions. Precisely, we extend the notions, introduced and thoroughly studied by R.T.~Rockafellar, of twice epi-differentiability and proto-differentiability to the case of a parameterized lower semi-continuous 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 proto-derivative 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 epi-derivatives 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 catching-up algorithm, we show the well-posedness (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 well-known 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, 831-843. DOI: 10.1007/s11590-017-1149-2. 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), 5--47.

• 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 well-known global optimization tools. These optimality conditions are new in the literature and are expressed in terms of max-min conditions with linked constraints.

• Prox-regularity approach to generalized equations and image projection, (with F. Nacry and L. Thibault). ESAIM: Control, Optimisation and Calculus of Variations 24, pp 677-708 (2018). DOI: cocv170112. Abstract

  Abstract: In this paper, we first investigate the prox-regularity behaviour of solution mappings to generalized equations. This study is realized through a nonconvex uniform Robinson-Ursescu type theorem. Then, we derive new significant results for the preservation of prox-regularity under various and usual set operations. The role and applications of prox-regularity of solution sets of generalized equations are illustrated with dynamical systems with constraints.

• Second-order State Dependent Sweeping Process with Unbounded and Nonconvex Constraints, (with B.K. Le). Pure and Applied Functional Analysis, no. 2, 271-285 (2018). . Abstract

  Abstract: In this paper, by using an implicit discrete scheme, we prove the existence of solutions for a class of second-order sweeping processes with possibly unbounded and prox-regular moving sets depending on both the time and the state in a Hilbert space. We use the local excess (local one-sided Lipschitz continuity property) instead of the global Hausdorff distance to describe the way the set of constraints is moving.

• Newton-type 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 341-370 (2018). Abstract

  Abstract: This paper is devoted to the study of Newton-type algorithm for solving inclusions involving set-valued 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 1017-1037. 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 semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for $a$-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) 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 Fenchel-Lagrange 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 regularized-scalarized 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.

• Quasi-Newton methods for solving nonsmooth equations: Generalized Dennis-Moré theorem and Broyden's update (with H.V. Ngai). Journal of Convex Analysis, 25 (2018), No. 4, pp 1075-1104. . Abstract

  Abstract: In this paper, we study the quasi-Newton method by using set-valued approximations for solving generalized equations without smoothness assumptions. The set-valued 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 Dennis-Mor\'{e} theorem, which gives a characterization of the $q-$superlinear convergence of the quasi-Newton 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 State-Dependent Sweeping Processes with Perturbations in Uniformly Convex and q-Uniformly Smooth Banach Spaces, (with B.K. Le). Numerical Algebra, Control and Optimization (NACO AIMS), Volume 8, Number 1, pp. 81-95 (2018). DOI: 10.3934/naco.2018005. Abstract

  Abstract: In this paper, the existence of solutions for a class of first and second order unbounded state-dependent 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 Fenchel-Lagrange 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 so-called Fenchel-Lagrange 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 Fenchel-Lagrange duality to the class of strong bilevel programming problems is new in the literature.

• Stability of metric regularity with set-valued perturbations and application to Newton's method for solving generalized equations, (with H. V. Ngai and N. Van Vu). Journal Set-Valued 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 set-valued perturbation. By adopting the measure of closeness between two multifunctions, we establish some stability results on the semi-local/local metric regularity. These results are applied to study the convergence of iterative schemes of Newton-type methods for solving generalized equations in which the set-valued 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 right-differentiable everywhere and its right-derivative is right-continuous. Nonsmooth Lyapunov pairs for this system are also analyzed.

• Maximal Monotonicity and Cyclic-Monotonicity 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 so-called 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.

• Strong-Weak Nonlinear Bilevel Problems: Existence of Solutions in a Sequential Setting, (with A. Aboussoror and F. Saissi). Journal Set-valued and Variational Analysis, 25 (2017), no. 1, 113–132. DOI 10.1007/s11228-016-0369-4 . Abstract

  Abstract: The paper deals with a strong-weak nonlinear bilevel problem $({S})$ which generalizes the well-known 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 123-135 (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 Fenchel-Lagrange 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 Prox-Regular Sets, (with F. Nacry and L. Thibault). ESAIM: Control, Optimisation and Calculus of Variations, Vol. 23, No 4 pp 1293-1329 (2017): https://doi.org/10.1051/cocv/2016053 . Abstract

  Abstract: In this paper, we study the well-posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular 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 catching-up 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 prox-regularity 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 prox-regularity of sets and application to constrained optimization, (with F. Nacry and L. Thibault). SIAM Journal on Optimization SIOPT, Vol. 26, No 1, pp 448-473 (2016) . Abstract

  Abstract: In this paper, we first provide counterexamples showing that sublevels of prox-regular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be prox-regular. Then, we prove the uniform prox-regularity of such sets under usual verifiable qualification conditions. The preservation of uniform prox-regularity of intersection and inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given.

• Implicit Euler Time-Discretization of a Class of Lagrangian Systems with Set-Valued Robust Controller, (with B. Brogliato and B.K. Le), Journal of Convex Analysis 23 (2016), No. 1, pp 23-52. Abstract

  Abstract: A class of Lagrangian continuous dynamical systems with set-valued controller and subjected to a perturbation force has been thoroughly studied in [S. Adly, B. Brogliato, B. K. Le, Well-posedness, robustness and stability analysis of a set-valued controller for Lagrangian systems, SIAM J. Control Optim., 51(2), 1592--1614, 2013]. In this paper, we study the time discretization of these set-valued systems with an implicit Euler scheme. Under some mild conditions, the well-posedness (existence and uniqueness of solutions) of the discrete-time 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 two-degree 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/s10957-016-0905-2 . 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 infinite-dimensional 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 Newton--type method for solving generalized equations involving set--valued maps in Banach spaces. Kantorovich--type 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 first-order 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.

• Finite-time stabilization and H_\infty Control of nonlinear time-varying 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 303-315. . Abstract

  Abstract: This paper studies the robust finite-time $H_\infty$ control for a class of nonlinear systems with time-varying delay and disturbances via ouput feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent 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 closed-loop system in the finite-time sense. An application to $H_\infty$ control of uncertain linear systems with interval time-varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.

• Optimal Guaranteed Cost Control of Nonlinear Time-Varying 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. 649-667 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 time-varying delays and nonlinear perturbations. By introducing a set of improved Lyapunov-Krasovskii functionals, a novel delay-dependent 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 LMI-based 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). Set-Valued and Variational Analysis, March 2016, Volume 24, Issue 1, pp 13-35. First online: 04 July 2015 . Abstract

  Abstract: In this paper, we study the well-posedness and stability analysis of set-valued Lur'e dynamical systems in infinite-dimensional Hilbert spaces. The existence and uniqueness results are established under the so-called 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 finite-dimensional 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 set-valued and variational analysis.

2015

• Newton's method for solving inclusions using set-valued 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 set-valued perturbations are presented. As an application, we study (super-)linear convergence of the Newton-type iterative process for solving generalized equations. The possibility to choose set-valued approximations allows us to describe several iterative schemes in a unified way (such as inexact Newton method, non-smooth Newton method for semi-smooth functions, inexact proximal point algorithm, etc.). Moreover, it also covers a forward-backward 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 Second-Order 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 one-sided Lipschitz stability of set-valued contractions, (with A.L. Dontchev and M. Théra), Numerical Functional Analysis and Optimization, 35 (2014), no. 7-9, 837–850 Abstract

  Abstract: We show that a result by T. -C. Lim [On fixed-point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436--441] 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 Lyusternik-Graves theorem is a corollary of this estimate as well. We apply the generalization of Lim's result to derive one-sided Lipschitz properties of the solution mapping of a differential inclusion with a parameter.

• Quantitative stability of a generalized equation. Application to non-regular 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 non-monotone generalized equation. The theoretical results are applied in the theory of non-regular 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. 1-2, Ser. B, pages: 5–47 Abstract

  Abstract: In this paper, we analyze and discuss the well-posedness of two new variants of the so-called 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 well-posedness 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 non-monotone set-valued Lur’e dynamical systems, (with B. K. Le) Applicable Analysis, 93 (2014), no. 5, 1087–1105. . Abstract

  Abstract: In this paper, we analyze the well-posedness, stability and invariance results for a class of non-monotone set-valued 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 set-valued 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 voltage-current characteristics are not monotone but only locally hypo-monotone. 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 hypo-monotonicity; 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 Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems, (with A. Aboussoror). Numerical Functional Analysis and Optimization, 35 (2014), no. 7-9, 816–836 Abstract

  Abstract: The paper deals with a generalized semi-infinite 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

• Well-posedness, Robustness and Stability Analysis of a Set-Valued 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 non-trivial mass matrix. First the existence and uniqueness of solutions are analyzed, then the Lyapunov stability, the Krasovskii-LaSalle invariance principle and finite-time convergence properties are studied.

• A New Method for Solving Pareto Eigenvalue Complementarity Problems, (with H. Rammal), Computational Optimization and Applications 55, No 3, pp 703-731 (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 bi-eigenvalue 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), Set-Valued and Variational Analysis 21 (2013), no. 2, 333–358. Abstract

  Abstract: The paper is devoted to the study of the Aubin/Lipschitz-like property and the isolated calmness of a particular non-monotone generalized equation arising in electronics. The variational and non-smooth analysis is applied in the theory of non-regular 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 non-smooth 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 1-2, pp 5-26 (2013) Abstract

  Abstract: We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic 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 Non-Monotone 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 non-monotone 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 non-monotone 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 DC-DC Buck Converter, (with D. Goeleven and B. K. Le), Set Valued and Variational Analysis (2012) 20:331-353 Abstract

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

• Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions, (with A.Hantoute and M. Thera), Nonlinear Analysis: Theory, Methods and Applications 75, 3 (2012) 985-1008 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 first-order 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 91-104 (2012). Abstract

  Abstract: We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic 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 Fenchel-Lagrange Duality Approach for a Bilevel Programming Problem with Extremal-Value Function, (with A. Aboussoror), Journal of Optimization Theory and Applications 149, 2 (2011) 254-268 Abstract

  Abstract: In this paper, for a bilevel programming problem (S) with an extremal-value function, we first give its Fenchel-Lagrange 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 one-level convex minimization problem.

• Finite-time Lyapunov stability analysis of evolution variational inequalities, (with K. Addi and H. Saoud), Discrete and Continuous Dynamical Systems - Series A 31, 4 (2011) 1023-1038 Abstract

  Abstract: Using Lyapunov's stability and LaSalle's invariance principle for nonsmooth dynamical systems, we establish some conditions for finite-time 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 cone-constrained eigenvalue problems, (with A. Seeger), Computational Optimization and Applications 49, 2 (2011) 299-318 Abstract

  Abstract: We study several variants of a nonsmooth Newton-type 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) 559-571 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 quasi-variational obstacle problem, (with M. Ait-Mansour and M. Bergounioux), Journal of Golobal Optimization, Vol. 47, Number 3, pp 421-435 (2010). Abstract

  Abstract: We consider an optimal control where the state-control relation is given by a quasi-variational 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 Mosco-convergence theory, that gives the weak closeness of the control-to-state 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 261-272 (2009).

2008

• A sensitivity analysis of a class of semi-coercive variational inequalities using recession tools, (with K. Addi, D. Goeleven and H. Saoud), Journal of Global Optimization, vol. 40, no. 1, pp. 7-27 (2008).

2007

• A method using the approach of Moreau and Panagiotopoulos for the mathematical formulation of non-regular circuits in electronics, (with K. Addi, B. Brogliato and D. Goeleven), Nonlinear Analysis: Hybrid Systems and Applications, N° 1, pp 315-324 (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 1-28 (2007).
• Stability analysis of filtered mass-spring systems, (with A. Ahmad, D. Ghazanfarpour and O. Terraz), Proceedings of theory and practice of computer graphics , 2007, pp 45-52 (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 629-650 (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 289-304, (2006).
• Attractivity Theory for Second Order Nonsmooth Dynamical Systems with application to dry friction, Journal of Mathematical Analysis and Applications , 322 pp 1055-1070 (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 quasi-variational inequalities, (with M. Ait-Mansour and L. Scrimali), Bulletino Della Unione Mathematica Italiana (BUMI) pp 767-772 (2006).
• Stability in frictional unilateral elasticity revisited : an application of the theory of semi-coercive variational inequalities, (with E. Ernst et M. Théra), Proceeding of American Institute of Physics, Vol. 835, pp 1-11 (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 471-478 (2006).
• Stability of linear semi-coercive variational inequalities in Hilbert spaces: application to the Signorini-Fichera Problem, Journal of Nonlinear and Convex Analysis, N° 3, pp 325-334 (2006). Abstract

  Abstract: In this paper we show how recent results concerning the stability of semi-coercive variational inequalities on reflexive Banach spaces, obtained in 25 can be applied to establish the stability of the semi-coercive Signorini-Fichera 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 315-324 (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 73-80 (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 2911-2915 (2004). Abstract

  Abstract: The aim of this note is to characterize the norm-closure 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.

• Well-positioned closed convex sets and well-positioned closed convex functions, (with E. Ernst and M. Théra), Journal of Global Optimization , 29 (4), pp 337-351 (2004).
• A stability Theory for second-order nonsmooth dynamical systems with application to friction problems, (with D. Goeleven), Journal de Mathématiques Pures et Appliquées , Vol. 83, pp 17-51 (2004). Abstract

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

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 1219-1249 (2003). Abstract

  Abstract: We characterize in a reflexive Banach space all the closed convex sets C₁ containing no lines for which the condition C₁^∞ ∩ C₂^∞ = ( 0 ); ensures the closedness of the algebraic difference C₁ - C₂ of all closed convex sets C₂. We also answer a closely related problem: determine all the pairs C₁ ,C₂ of closed convex sets containing no lines such that the algebraic difference of any sufficiently small uniform perturbations of C₁ and C₂ 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 145-160 (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 34-39 (2002).

2001

• A Characterization of Convex and Semicoercive Functionals, (with E. Ernst and M. Théra) Journal of Convex Analysis , Volume 8, No 1, pp 127-148 (2001). Abstract

  Abstract: In this paper we prove that, if every small L^∞-perturbation of a proper, convex, l.s.c., extended-real-valued functional defined on a reflexive Banach space attains its minimum value, then this functional is semi-coercive.

• 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 409-414 (2001).

2000

• A Discretization Theory for a Class Semi-coercive Unilateral Problems, (with D. Goeleven), Numerishe Mathematik , 87, pp 1-34 (2000). Abstract

  Abstract: In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a Signorini-Fichera 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 non-convex superpotentials, (with D. Motreanu) Journal of Global Optimization, 17, pp 9-17 (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 variational-hemivariational inequalities, (avec D. Motreanu), Journal of Nonlinear and Convex analysis , Vol. 1, No 3, pp 255-270, (2000).

1999

• Solvability of generalized nonlinear symmetric variational inequalities, ( with W. Oettli), Bullettin of the Austrelian Mathematical Society, Ser. B (40), pp 289-300 (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. 711-718 (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 11--26 (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 so-called 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 stress-strain or force-displacement (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 semi-permeable problems, adhesive grasping problems in robotics, loading-unloading 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 1573-1603 (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 Navier-Stockes 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, Navier-Stokes equations, puffed-up membrane, elastic plate,unilateral buckling.

• A perturbed iterative method for a general class of variational inequalities, Serdica Mathematical Journal , Vol. 22, pp 69-82 (1996). Abstract

  Abstract: The generalized Wiener-Hopf 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 609-630 (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 47-58 (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 well-developed theory in mathematics. The mechanical meaning of a variational inequality is given by the formulation of the principle of virtual work when a monotone stress-strain or reaction-displacement condition hold. In the case of lack of monotonicity of these underlying stress-strain or reaction-displacement 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 229-240 (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. 397-409, 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.