Samir Adly

University of Limoges | UNILIM · Department of Mathematics and Computer Sciences

In this paper, we study a sliding mode observer for a class of set-valued Lur'e systems subject to uncertainties. We show that our approach has obvious advantages than the existing Luenberger-like observers. Furthermore, we provide an effective continuous approximation to eliminate the chattering effect in the sliding mode technique.

In this paper, we propose to solve Pareto eigenvalue complementarity problems by using interior-point methods. Precisely, we focus the study on an adaptation of the Mehrotra Predictor Corrector Method (MPCM) and a Non-Parametric Interior Point Method (NPIPM). We compare these two methods with two alternative methods, namely the Lattice Projection M...

We compare in this note a variety of methods for solving inverse Pareto eigenvalue problems which are aimed at constructing matrices whose Pareto spectrum contains a prescribed set of distinct reals. We choose to deal with such problems by first formulating them as nonlinear systems of equations which can be smooth or nonsmooth, depending on the ch...

In a Hilbert space setting, this paper is devoted to the study of a class of first-order algorithms which aim to solve structured monotone equations involving the sum of potential and nonpotential operators. Precisely, we are looking for the zeros of an operator A=∇f+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepacka...

The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality o...

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, w...

In this paper we provide a Pontryagin maximum principle for optimal sampled-data control problems with nonsmooth Mayer cost function. Our investigation leads us to consider, in a first place, a general issue on convex sets separation. Precisely, thanks to the classical Fan's minimax theorem, we establish the existence of a universal separating vect...

In a Hilbert space $\mathcal{H}$ H , we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator $A= \nabla f +B $ A = ∇ f + B , where ∇ f is the gradient of a continuously differentiable convex fu...

In this paper, we provide a new application of the Douglas–Rachford splitting method for finding a zero of the sum of two maximal monotone operators where one of them depends implicitly on the state variable. Our proposed algorithms are much simpler with better rate of convergence than existing results and can be implemented under general condition...

In a Hilbert space \( {\mathcal H}\), we introduce a new class of first-order algorithms which naturally occur as discrete temporal versions of an inertial differential inclusion jointly involving viscous friction and dry friction. The function \(f:{\mathcal H}\rightarrow {\mathbb {R}}\) to be minimized is supposed to be differentiable (not necessa...

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

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

This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic support functionals and points of a given set, all associated to a positive constant r. By means of these concepts we obtain nonlinear functional separations for points and pro...

We give new criteria for weak and strong invariant closed sets for differential inclusions in \(\mathbb {R}^{n}\), and which are simultaneously governed by Lipschitz Cusco mapping and by maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions and pairs. The resulting condition...

This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally 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 o...

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...

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 dist...

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 frict...

Dry friction gives finite convergence properties for inertial proximal-gradient algorithms

In this paper, we provide a new application of the Douglas-Rachford splitting method for finding a zero of the sum of two maximal monotone operators where one of them depends implicitly on the state variable. Our proposed algorithms are much simpler with better rate of convergence than existing results and can be implemented under general condition...

The aim of the present work is to provide an explicit decomposition formula for the resolvent operator \(\mathrm {J}_{A+B}\) 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 \(\mathrm {J}^A_B\), which generalizes the usual notion. Then, ou...

In this paper, we study the existence and the stability in the sense of Lyapunov of differential inclusions governed by the normal cone to a given prox-regular set, which is subject to a Lipschitzian perturbation. We prove that such apparently more general non-smooth dynamics can be indeed remodeled into the classical theory of differential inclusi...

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint C ( ⋅ , u ) C(⋅,u) depends upon the unknown state u 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...

We present an extended conjugate duality for a generalized semi-infinite programming problem (P). The extended duality is defined in the context of the absence of convexity of problem (P), by means of a decomposition into a family of convex subproblems and a conjugate dualization of the subproblems. Under appropriate assumptions, we establish stron...

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 mov...

The main concern of this paper is to investigate sensitivity properties of parametric evolution systems of first 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 subsmooth or semiconvex functions, we give some c...

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 t...

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...

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 cons...

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...

We first give criteria for weak Lyapunov functions associated to differential inclusions, given in ℝ ⁿ and governed by Lipschitz Cusco perturbations of maximal monotone operators. Next, we apply this result to study the existence and stability of differential inclusions involving normal cones to prox-regular sets.

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 (Adly et al. in Math Program Ser B 148(1):5–47, 2014). Our aim is to improve Adly et al. (2014, Theorem 5.1) to allow possibly unbounded m...

In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of 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 evolut...

In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of 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 evolut...

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, a...

In this paper we are interested in a strong bilevel programming problem (S). For such a problem, we establish necessary and sufficient global optimality conditions. Our investigation is based on the use of a regularization of problem (S) and some well-known global optimization tools. These optimality conditions are new in the literature and are exp...

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...

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...

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 clas...

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 f-proximal operator of g and denoted by proxfg, that generalizes the classical notion. Then we prove the de...

In this paper, we present a duality approach using conjugacy for a semivectorial bilevel programming problem (S) where the upper and lower levels are vectorial and scalar respectively. This approach uses the Fenchel-Lagrange duality and is given via a regularization of problem (S) and the operation of scalarization. Then, using this approach we pro...

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-ty...

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 o...

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-r...

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 differen...

The main result of this paper provides an explicit decomposition of the proximity operator of the sum of two proper, lower semicontinuous and convex functions. For this purpose, we introduce a new operator, called f-proximity operator, generalizing the classical notion. After providing some properties and characterizations, we discuss the relations...

By using a regularization method, we study in this paper the global existence and uniqueness property of a new variant of non-convex 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...

We study the existence and stability of solutions for\ differential inclusions governed by the normal cone to a prox-regular set and subject to a Lipschitz perturbation. We prove, that such apparently more general systems can be indeed remodeled into the classical theory of differential inclusions involving maximal monotone operators. This surprisi...

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 as in , while the invariance criteria are still written...

The paper deals with a strong-weak nonlinear bilevel problem which generalizes the well-known weak and strong ones. In general, the study of the existence of solutions to such a problem is a difficult task. So that, for a strong-weak nonlinear bilevel prob- lem, we first give a regularization based on the use of strict ε-solutions of the lower leve...

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 the regularized problem of (S) admits solutions and any accumulation point of a sequence of regularized solutions solves (...

This chapter focuses on Moreau’s sweeping processes. Existence and uniqueness results are given when the moving set of constraints is assumed to be convex and absolutely continuous or has a bounded retraction. A new variant of Moreau’s sweeping process with velocity constraint in the moving set is also analyzed. Some applications of the sweeping pr...

This chapter is dedicated to the study of Lur’e systems involving maximal monotone and nonmonotone set-valued nonlinearities. The first case is studied with a nonzero feedthrough (or feedforward) matrix D under the so-called passivity condition. In the second case, the matrix D=0 and the problem are formulated into a first-order differential inclus...

The aim of this chapter is to study the existence and uniqueness of solution to a first-order nonsmooth dynamical system involving the subdifferential of a convex, lower semicontinuous and proper function. These problems are also known as evolution variational inequalities. Some conditions ensuring the stability, the asymptotic stability and the fi...

The main purpose of this chapter is to provide the reader with the knowledge of some basic concepts in convex analysis, nonsmooth analysis and Lyapunov stability theory. Useful mathematical results throughout the book are discussed (without proofs) and illustrated with figures.

This chapter provides a mathematical theory applicable to the study of second-order dynamic systems with unilateral contact and friction. Conditions ensuring stability (in the sense of Lyapunov), attractivity, and asmptotic stability are given. Some illustrative small-sized examples in unilateral mechanics and in nonregular electrical circuits are...

In this chapter, an overview of some mathematical models in nonsmooth dynamics is given. The main purpose is to give the reader a quick but comprehensive snapshot of other classes of nonsmooth systems that can/cannot be captured by the models studied in detail in this book. The following are reviewed: the piecewise dynamical systems; the Filippov c...

Applications in unilateral mechanics and electronics, With a foreword by J.-B. Hiriart-Urruty

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 condi...

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...

The guaranteed cost control problem via static output feedback controller for a class of nonlinear systems with interval and non-differentiable time-varying delays is investigated. By constructing a set of improved Lyapunov-Krasovskii functionals, a novel delay-dependent condition for designing output feedback controllers with guaranteed exponentia...

In this paper, an existence and uniqueness result of a class of second-order sweeping processes, 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 possibly unbounded. The assumptions...

A class of Lagrangian continuous dynamical systems with set-valued controller and subjectedto a perturbation force has been thoroughly studied in [3]. In this paper, we study the timediscretization of these set-valued systems with an implicit Euler scheme. Under some mildconditions, the well-posedness (existence and uniqueness of solutions) of the...

In this paper, we study Newton-type methods 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 Smale's classical –theory to generalized equations. These results are new and can...

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 interse...

The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in the previous paper (Adly et al. in Nonlinear Anal 75(3): 985–1008, 2012). This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential i...

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...

This paper studies the robust finite-time Ho., control for a class of nonlinear systems with time-varying delay and disturbances via output 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 line...

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 a Newton-type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton's method, the nonsmooth Newton's method for semi...

In this paper, we analyze and discuss the well-posedness of two newvariants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971) 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), suppos...

We give conditions under which the distance from a point x to the set of fixed points of the composition of the set-valued mappings F and G is bounded by a constant times the smallest distance between F −1(x) and G(x). This estimate allows us to significantly sharpen a result by T.-C. Lim [1010. T.-C. Lim ( 1985 ). On fixed-point stability for set...

This article 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...

In this paper, we study numerical methods for solving eigenvalue complementarity problems involving the product of second-order cones (or Lorentz cones). We reformulate such problem to find the roots of a semismooth function. An extension of the Lattice Projection Method (LPM) to solve the second-order cone eigenvalue complementarity problem is pro...

The paper is devoted to the study of several stability properties (such as Aubin/Lipschitz-like 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 diode alter...

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 LP...

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 uniquene...

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, silico...

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 L...

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 Krasovskii-LaSalle invariance principle, and finite-time convergence properties are studied.

In this paper, we analyse 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. Many recent researches deal with the case when the set-valued part is the sub-differential of some proper, convex, lower semicontinuous functio...

The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated cahnness) 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...

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

We study 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 res...

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 mean...

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.