| Titre : | Asymptotic Behavior of Solutions of Some Viscoelastic Problems |
| Auteurs : | Soumia Hamdi, Auteur ; Amirouche Berkani, Directeur de thèse |
| Type de document : | Thése doctorat |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2024 |
| Format : | 1 vol. (119 p.) / couv. ill. en coul / 30 cm |
| Langues: | Anglais |
| Mots-clés: | PDEs ; dynamic boundary condition ; Euler-Bernoulli beam ; existence and uniqueness of solutions ; Galerkin approximation method ; arbitrary decay ; viscoelasticity ; relaxation function |
| Résumé : |
The main purpose of this thesis is to investigate the existence and uniqueness of solutions, as well as the asymptotic behavior of some viscoelastic problems in one-dimensional space, precisely, this work addresses the problem of undesirable vibrations and the control of attitude stabilization of a flexible satellite during the maneuvers. In view of this, viscoelastic materials are suggested to attenuate or suppress the unwanted vibrations of a flexible satellite. The flexible satellite system consists of a central rigid hub and two large symmetric flexible appendages. Mathematically, the problem can be modeled by a set of partial differential equations (PDEs) taking into account the dynamic boundary condition. Our research utilizes Lyapunov’s direct method to study some viscoelastic systems. The results obtained in this thesis aim to enhance much of the previous scientific research. |
| Sommaire : |
Dedication ii Acknowledgement iii Abstract iv Résumé v Introduction 1 0.1 The main objective of this wor . . 8 0.2 Outline of the thesis . . . . . . . . . . .. . . 8 1 Preliminary results 10 1.1 Basic notions in viscoelasticity linear . .. . . . 10 1.1.1 Viscoelastic materials . . . . . . . .. . 10 1.1.2 Hooke’s law . . . . . . . . . . . 12 1.1.3 Linear viscoelasticity . . . . . . . . . . . . . . . 12 1.1.4 Fundamental uniaxial tests in one-dimensional viscoelastic behavior 12 1.1.5 Solids and liquids . . . . . . . . . . . . 16 1.1.6 Behavior law of linear viscoelastic material . . . . . 16 1.1.7 Boltzmann superposition principle . . . . . . . . . 17 1.1.8 Viscoelastic models . . . . . . . . . . . . . . . . 18 1.2 Reminders in functional analysis . . . . . . . . . . . . . . . 21 1.2.1 Weak topology and weak-star topology . . .. . . 22 1.2.2 Ck(Ω) Spaces . . . . . . . . . . . . . . . . . . . . . . . 24 1.2.3 Reflexive spaces . . . . . . . . 25 vi1.2.4 Separable spaces . . . . . . . . . . . . . . . . 25 1.2.5 Definitions and elementary properties of Lp(Ω) spac. . 25 1.2.6 Definition and elementary properties of Sobolev spaces . . . . . 26 1.2.7 Spaces of vector-valued functions . . . . . 30 1.2.8 Duality and weak convergence . . . . . . . . . . 31 1.2.9 Faedo-Galerkin’s approximations . . . . . . . . 32 1.2.10 Aubin-Lions lemma . . . . . . . . . . . . . 33 1.2.11 Basic inequalities . . . . . . . . . . . . . . 34 2 Arbitrary decay of solutions for a viscoelastic flexible satellite system 37 2.1 Introduction . . . . . . . . . . . .. . . 37 2.2 Preliminary results . . . . . . . . . . . . . . . . 39 2.3 Technical lemmas . . . . . . . . . . . . . . . . 42 2.4 Asymptotic behavior . . . . . . . . . . . . . .. 54 3 Existence and uniform decay for a flexible satellite system under unknown distributed disturbance63 3.1 Introduction . . . . . . . . 63 3.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Well posedness . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Technical lemmas . . . . . . . . . . . . . 75 3.5 Asymptotic behavior . . . . . . . . . . . . . . . . . 85 4 Stabilization of a flexible satellite system with memory term under tension effects 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Preliminary Results . . . . . . . . . . . . . . . . . . . 93 4.3 Technical lemmas . . . . . . . . . . . . . . . . . . . 99 4.4 Asymptotic behavior . . . . . . . . . . . . . . . . .. . 109 Appendix 113 4.5 Introduction . . . . . . . . . . . . . . . . . . . . . 114 4.6 Mathematical modeling of a flexible satellite . . . . . . . 116 viiTable of Contents Conclusion 118 Bibliographies 119 |
| En ligne : | http://thesis.univ-biskra.dz/id/eprint/6571 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| TM/162 | Théses de doctorat | bibliothèque sciences exactes | Consultable |
