Prof. REZZOUG Imad | University Of Oum El Bouaghi (original) (raw)

Books by Prof. REZZOUG Imad

Cours : Méthodes Numériques, 2020

Ce document propose aux étudiants de niveau troisième année de Licence en mathématiques et premiè... more Ce document propose aux étudiants de niveau troisième année de Licence en mathématiques et première année de Master en mathématiques appliquées, contient des cours détaillés. Il combine d'une façon indissociable l'étude des équations aux dérivées partielles avec les méthodes numériques des éléments finis et des différences finies. C'est un cours illustré par des exemples aléatoires et des exercices se trouvent à la fin de ce polycopié avec plusieurs réponses possibles que des exercices à étapes et utilisant des bases de données importantes.

Ce document propose aux étudiants de niveau deuxième année de Licence en mathématiques, contient ... more Ce document propose aux étudiants de niveau deuxième année de Licence en mathématiques, contient des cours détaillés. Il combine d'une façon indissociable l'étude des concepts bilinéaires des formes quadratiques avec l'étude matricielle. C'est un cours illustré par des exemples aléatoires et des exercices se trouvent à la fin de chaque chapitre avec plusieurs réponses possibles que des exercices à étapes et utilisant des bases de données importantes.

La problématique de ce travail était détudier la contrôlabilité, par un nombre réduit de contrôl... more La problématique de ce travail était détudier la contrôlabilité, par un nombre réduit de contrôles, de certaines classes de systèmes gouvernés par des équations aux dérivées partielles. Donc nous donnons les diverses caractérisations liées à la notion de la contrôlabilité en dimension fini et technique de résolution des problèmes dévolution par la théorie des semi-groupes. Enfin, on a présenté une nouvelle façon daborder la contrôlabilité faible pour des systèmes paraboliques linéaires, puis dutiliser la dualité pour formuler la théorie dobservabilité correspondante.

Teaching Documents by Prof. REZZOUG Imad

Séries des Travaux Dirigés et Sujets D'examens et Rattrapages avec Corriges pour les étudiants en... more Séries des Travaux Dirigés et Sujets D'examens et Rattrapages avec Corriges pour les étudiants en économie.

Ce cours est une enquête de pratique des techniques de résolution numérique des différentes class... more Ce cours est une enquête de pratique des techniques de résolution numérique des différentes classes d’équations aux dérivées partielles (elliptiques, paraboliques et hyperboliques). L’accent sera mis sur la programmation de schémas numériques à des problèmes pratiques dans l’ingénierie et les sciences physiques. La résolution comprendra la méthode des différences finies. Pratiquement, elles seront pleinement utilisées en MATLAB et ses fonctionnalités de programmation.

Thesis Chapters by Prof. REZZOUG Imad

Memory of master , 2024

In this memory, we study the problem of optimal control for an electric vehicle, by minimizing th... more In this memory, we study the problem of optimal control for an electric vehicle, by minimizing the energy consumed with a direct method which consists of transforming an optimal control problem into a nonlinear optimization problem in finite dimension by discretizing the status and control. This type of problem is generally modeled using localized controlled systems. We then applied it to our model by carrying out simulations in Matlab to illustrate the results. Imposing constraints on speed can lead to either an increase or a decrease in energy consumption.

Mémoire de Master, 2024

Identification problems consist of finding the causes of a phenomenon based on observation, and r... more Identification problems consist of finding the causes of a phenomenon based on observation, and resolving this type of problem is done through experimental measurement. In this work, we propose to study a class of problems governed by a parabolic equation with incomplete data. For this purpose, we use the concept of sentinels introduced by J. L. Lions, which is the most commonly used strategy, involving obtaining information about the causes from a weighted average of the observation. We then prove the existence of the average sentinel function by solving a zero-average controllability problem with control constraints. Next, we are concerned with identifying pollution terms present in the heat equation with missing data.
Keywords : Average Sentinel; Identification Method; Average Observability; Pollution Term.
2020 Mathematics Subject Classification : Primary 05C38, 15A15 ; Secondary 05A15, 15A18.

Mémoire de Master, 2023

Our work aims to study the optimal control problem of localized and distributed systems, two case... more Our work aims to study the optimal control problem of localized and distributed systems, two cases will be presented :
The first case deals with the control applied on the inverted pendulum equation. The necessary condition of optimality is deduced from Pontryagin's maximum principle, while the simple or inverted pendulum problem is resolved with Runge Kutta's method. The numerical results are presented in MATLAB.
The second case studies the exact controllability of the wave equation and the control applied on the boundary. The method used is the Hilbert uniqueness method "HUM method" introduced by J.L. Lions. Thismethod is based on uniqueness criterion and energy inequalities that characterize the space of the initial data.

Keywords : Inverted pendulum, Wave equation, HUM method, Controllability, Optimal control, Stability.
2010 Mathematics Subject Classification : 35L05, 93B05, 49J15, 49J20, 49J25, 34D20, 93D20.

Dans ce travail, on s'intéresse à étudier d'un autre type de sentinelle appelé sentinelle approch... more Dans ce travail, on s'intéresse à étudier d'un autre type de sentinelle appelé sentinelle approchée ou faible pour identifier des paramètres de pollution aux des systèmes distribuées, on construit les sentinelles approchées à partir de l'existence de la contrôlabilité approchée d'un système adjoint. Finalement le développement de Taylor du premier ordre de la sentinelle donne l'identification de terme de pollution.
Mots-clés : Système distribué, contrôlabilité, terme de pollution, méthode de Sentinelle, données manquantes.

Dans ce mémoire, nous essayons d'identifier le terme de pollution dans le problème de Petrowsky q... more Dans ce mémoire, nous essayons d'identifier le terme de pollution dans le problème de Petrowsky qui est constitué dans la condition de la frontière, et les termes manquants sont des outils utilisés. Pour résoudre le problème d'identification, on utilise la méthode de la sentinelle de J.L. Lions. Le terme de pollution est estimé par le développement Taylor de cette méthode.
Mots-clés : Contrôlabilité, Semi groupe, Sentinelle, Système de Petrowsky.

The objective of this memory is to study the optimal control of distributed systems with incomple... more The objective of this memory is to study the optimal control of distributed systems with incomplete data. Particularly control for the heat equation with missing initial condition and delay parameter. The low-regret control seems to be the best method to solve this kind of problems which characterized by an optimality systems.

Mémoire de Master, 2021

La théorie des semi-groupes intégrés a de très vastes applications allant des équations différent... more La théorie des semi-groupes intégrés a de très vastes applications allant des équations différentielles à retard, équations paraboliques avec retard ou encore avec des conditions de bords non linéaires, non locales et éventuellement non bornées, ou encore des équations structurées en âge (avec retard, diffusion,…).
Ainsi, l’utilisation des semi-groupes intégrés permet d’obtenir des résultats relativement fins sur la dynamique de certaines équations d’évolution qui n’entrent pas dans le cadre de la théorie classique des semi-groupes.

Mémoire de Master, 2021

Dans ce travail, on cherche le contrôle optimal par le principe du maximum de Pontryagin qui perm... more Dans ce travail, on cherche le contrôle optimal par le principe du maximum de Pontryagin qui permet d'écrire de façon pratique les conditions d'optimalités associe à un problème du contrôle optimal faisant intervenir la solution d'une équation différentielle contrôlée.

L'objectif de ce mémoire est d'étudier comment calculer numériquement le contrôle optimal appliqu... more L'objectif de ce mémoire est d'étudier comment calculer numériquement le contrôle optimal appliqué à l'équation de chaleur en utilisant la méthode des différences finies.

Ce travail concerne l’étude d’une classe de problèmes de Cauchy mal-posés, associe à un opérateur... more Ce travail concerne l’étude d’une classe de problèmes de Cauchy mal-posés, associe à un opérateur linéaire A de domaine dense définie dans un espace de Banach X. On montre que si -A est le générateur d'un semi-groupe analytique, alors il existe une famille régularisant d'opérateurs pour le problème en question. La méthode d'étude est basée sur celle de quasi réversibilité en utilisant les puissances fractionnaires et les semi-groupes d'opérateurs linéaires.

The objective of this thesis is to study the approximate controllability with constraints on the ... more The objective of this thesis is to study the approximate controllability with constraints on the control of the heat equation. To demonstrate the approximate controllability, it suffices to use Hahn Banach theorem and notice that the orthogonal of the set of reachable states at the moment is reduced to zero. Finally we adapted the method of duality (primal dual) of Fenchel-Rockafellar to characterize the optimal control.

L’objectif de ce mémoire est d’étudier la contrôlabilité des systèmes linéaires à énergie constan... more L’objectif de ce mémoire est d’étudier la contrôlabilité des systèmes linéaires à énergie constante (conservatifs) en dimension infinie. Pour atteindre cet objectif, nous avons adopté le critère de Hautus.
Mots clés: Systèmes linéaires, Contrôlabilité, Critère de Hautus.

L’objectif de cette thèse est d’étudier la contrôlabilité approchée avec contrainte sur le contrô... more L’objectif de cette thèse est d’étudier la contrôlabilité approchée avec contrainte sur le contrôle de l’équation de la chaleur et en suite nous
adaptons la méthode de Fenchel Pockafellar (voir [13]) pour caractériser le contrôle optimal.
Mots clés: Équation de la chaleur, Contrôlabilité approchée, Contrôle optimal.

Dans ce travail, on s’intéresse à l’application d’un autre type de sentinelle appelé sentinelle f... more Dans ce travail, on s’intéresse à l’application d’un autre type de sentinelle appelé sentinelle frontière pour identifier des paramètres de pollution présents sur le bord d’un lac. Ces sentinelles ont été introduites par Lions et ensuite généralisées par Nakoulima dans le cas où le domaine du contrôle est disjoint de celui de l’observation.
En se basant sur cette généralisation, on construit les sentinelles frontières à partir de l’existence de la contrôlabilité frontière d’un système adjoint. Enfin, on étudie l’existence et la construction des sentinelles frontières généralisées ayant une certaine sensibilité pour une observation sans bruit.

Papers by Prof. REZZOUG Imad

WSEAS transactions on systems and control/Wseas transactions on systems and control, Dec 31, 2023

Sentinel method introduced in the study of problems with incomplete data, particularly in the con... more Sentinel method introduced in the study of problems with incomplete data, particularly in the context of distributed systems where pollution terms may arise at the boundary. The idea of sentinel likely involves constructing a surrogate or placeholder value that helps account for missing data or uncertainties in the system. Weakly sentinel appears to be a modification or extension of the concept of a sentinel specifically tailored for estimating pollution terms in distributed systems with missing data. The term weakly might suggest that this sentinel is not as robust or precise as the ideal sentinel, but it serves a similar purpose in providing estimates or approximations in situations where complete data is not available.

Cours : Méthodes Numériques, 2020

Memory of master , 2024

Mémoire de Master, 2024

Mémoire de Master, 2023

Keywords : Inverted pendulum, Wave equation, HUM method, Controllability, Optimal control, Stability.
2010 Mathematics Subject Classification : 35L05, 93B05, 49J15, 49J20, 49J25, 34D20, 93D20.

Mémoire de Master, 2021

WSEAS transactions on systems and control/Wseas transactions on systems and control, Dec 31, 2023

$Research paper thumbnail of Inverse coefficient super-linear problem for a time fractional parabolic equation under integral overdetermination conditions$

Mathematica Montisnigri, 2023

In this paper, we investigate the inverse problem of the determining the righthand side of a nonl... more In this paper, we investigate the inverse problem of the determining the righthand side of a nonlinear fractional parabolic equation with integral overdetermination condition. For the solvability of the direct problem we use the energy inequality method also known as the methode of functional analysis or the method of a priori estimates, this method has a higher character in that the existence theorem can be derived from the solution of the problem posed, begining with the uniqueness theorem. The most difficult aspect of this method is selecting the functional spaces E and F and for the unique solvability of the iverse problem is estblished under suitable conditions. The fixed point theorem is used to establish the existence and uniqueness of the solution to the inverse problem based on the data, this inverse problem appears frequently in engineering and physics modeling of various phenomena. Sismology, medicine, continuous casting, fusion welding, oil and gas production during drilling and well operation.

$Research paper thumbnail of A study of a high-order time-fractional partial differential equation with purely integral boundary conditions$

Fractional differential calculus, 2023

The aim of this paper is to investigate the existence and uniqueness of the strong solution for t... more The aim of this paper is to investigate the existence and uniqueness of the strong solution for the linear time-fractional partial differential equation with purely integral conditions. The aimed investigation is demonstrated based on the so-called energy inequality method and the density of the operator generated by the considered problem. To do so, we first set the position of the problem under consideration coupled with its corresponding equivalent problem, say problem (x). Afterward, we introduce some necessary functional spaces needed for exploring the existence and uniqueness of solution of problem (x). Finally, we investigate the existence and uniqueness of solution of the main operational equation.

Journal of the Nigerian Society of Physical Sciences

This paper is interested in studying the one-phase Stefan problem. For this purpose, we use the n... more This paper is interested in studying the one-phase Stefan problem. For this purpose, we use the nonlinear sentinel method, which relies typically on the approximate controllability and the Fanchel-Rockafellar duality of the minimization problem, to prove the existence and uniqueness of a solution to this problem. In particular, our research focuses on the application of the nonlinear sentinel method to the single-phase Stefan problem. This approach aids in identifying an unspecified boundary section within the domain undergoing a liquid-solid phase transition. We track the evolution of the temperature profile in the liquid-solid material and the corresponding movement of its interface over time. Eventually, the local convergence used for the iterative numerical scheme is demonstrated.

WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL

We present in our work the steps of the Hilbert Uniqueness Method (HUM) and characterization of t... more We present in our work the steps of the Hilbert Uniqueness Method (HUM) and characterization of the low regret and no regret control. At the end of this article, we compare these methods.

In this work one interests a method for the identification of a part of the boundary in a parabol... more In this work one interests a method for the identification of a part of the boundary in a parabolic equation (Navier-Stokes equations). By the means of the controllability of an adjoint system one has to identify this part while basing on an observation made on part of the boundary known.

WSEAS transactions on systems, Mar 23, 2023

In this paper and in the first part of it, homotopy perturbation method is applied to solve secon... more In this paper and in the first part of it, homotopy perturbation method is applied to solve second order differential equation with non-constant coefficients. The method yields solutions in convergent series forms with easily computable terms (the convergence of this series is demonstrated in this paper). The result shows that this method is very convenient and can be applied to large class of problems. As for the second part, we found a solution of Telegraph equation using the Laplace transform and Stehfest algorithm method. Next, we used method of Homotopy perturbation. Finally, we give some examples for illustration.

Zenodo (CERN European Organization for Nuclear Research), Apr 27, 2021

In this paper, we will study practical problems of optimal control governed by differential equat... more In this paper, we will study practical problems of optimal control governed by differential equations with partial derivatives. In the framework of this work, we will deal with an optimal control problem applied to the heat equation. This problem has many applications in the field of physics and in automation.

WSEAS transactions on systems and control, Dec 9, 2022

The objective of this paper is to study the optimal control of distributed systems with incomplet... more The objective of this paper is to study the optimal control of distributed systems with incomplete data. Particularly control for the heat equation with missing initial condition and delay parameter. The low-regret control seems to be the best method to solve this kind of problems which characterized by an optimality systems.

WSEAS TRANSACTIONS ON SYSTEMS

WSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS

We revisit the notion of discrete control by M. Asch considering the case where the control is di... more We revisit the notion of discrete control by M. Asch considering the case where the control is discreet. This point of view leads to problems of controllability with or without constraints on the control. This article focuses on the case of hyperbolic equations although similar developments can be done for other PDE’s. The main tool used is the HUM method which is "adapted" to the constraints.

WSEAS TRANSACTIONS ON SYSTEMS AND CONTROL

$Research paper thumbnail of Solvability of a solution and controllability of partial fractional differential systems$

Journal of Interdisciplinary Mathematics

Abstract The paper is devoted to study the solvability of a solution for a Dirichlet problem for ... more Abstract The paper is devoted to study the solvability of a solution for a Dirichlet problem for a class of time-fractional partial parabolic differential system, by using the energy inequality method for the primal problem. Then, by the Tikhonov regularization method, we can study the approximate controllability of the dual problem. We were therefore able to study the solvability of a solution and controlability of this problem.

$Research paper thumbnail of Solvability of solution of singular and degenerate fractional nonlinear parabolic Dirichlet problems$

Bulletin of the Institute of Mathematics Academia Sinica NEW SERIES

In this paper, we establish the existence and uniqueness of the weak solution in functional weigh... more In this paper, we establish the existence and uniqueness of the weak solution in functional weighted Sobolev space for a class of initial-boundary value degenerate and singular fractional semi-linear parabolic problems. The results are established by using a priori estimate and applying an iterative process based on results obtained for the linear problem.

Innovative Journal of Mathematics (IJM)

In this paper, we consider a wave equation with pollution term that includes a fractional derivat... more In this paper, we consider a wave equation with pollution term that includes a fractional derivative of Riemann- Liouville with missing Newman boundary condition and the operator depending on unknown parameter. First we try to avoid the unknown parameter and to isolate the missing function the results are established by introduce the notion of average no regret sentinel. Second is applied the classical approach of sentinel to identify the pollution.

Innovative Journal of Mathematics (IJM)

In this work, we study an approximate controllability problem with constraint on the control. Thi... more In this work, we study an approximate controllability problem with constraint on the control. This problem appears naturally of weak and pointwise sentinel with a small or singleton region observation. The main tool is a theorem of uniqueness of the solution of ill-posed Cauchy problem for the parabolic equation.

Proyecciones (Antofagasta), 2021

The aim of this paper is about presenting some results of the Sentinel Theory in Connection with ... more The aim of this paper is about presenting some results of the Sentinel Theory in Connection with Control Theory of Distributed Systems. There is of course a large variety of models where the results to follow could be applied. What we have particularly in mind is the classical set of Navier-Stokes equations. We shall denote here by the velocity field and the pressure, which is a quite unusual notation in the "Turbulence" circle. The reason is simply that in all what is following that we think of as the state of our system, this state depends on control functions, these control functions are being either "artificial" or "natural". Also in this work we are(trying) working to get instantaneous information at fixed instant "T" on pollution term in Navier-Stokes system in which the initial condition is incomplete. The best method which can solve this problem is the sentinel one; it allows the estimation of the pollution term at which we look for in...