Notice bibliographique
Résumé
Feedback control is used in almost every aspect of modern life and is essential in almost all engineering systems. Since no mathematical model is perfect and disturbances occur frequently, feedback is required. The design of a feedback control has been widely investigated in finite-dimensional space. However, many systems of interest, such as fluid flow and large structural vibrations are described by nonlinear partial differential equations and their state evolves on an infinite-dimensional Hilbert space. Developing controller design methods for nonlinear infinite-dimensional systems is not trivial. \n \nThe objectives of this thesis are divided into multiple tasks. First, the well-posedness of some classes of nonlinear partial differential equations defined on a Hilbert space are investigated. The following nonlinear affine system defined on the Hilbert space H is considered \nz ̇(t)=F(z(t))+Bu(t), t≥0 z (0) = z0, \nwhere z(t) ∈ H is the state vector and z0 is the initial condition. The vector u(t) ∈ U, where U is a Hilbert space, is a state-feedback control. The nonlinear operator F : D ⊂ H → H is densely defined in H and the linear operator B : U → H is a linear bounded operator. Conditions for the closed-loop system to have a unique solution in the Hilbert space H are given. \n \nNext, finding a single bounded state-feedback control for nonlinear partial differential equations is discussed. In particular, Lyapunov-indirect method is considered to control nonlinear infinite-dimensional systems and conditions on when this method achieves the goal of local asymptotic stabilization of the nonlinear infinite-dimensional system are given. \n \nThe Kuramoto-Sivashinsky (KS) equation defined in the Hilbert space L2(−π,π) with periodic boundary conditions is considered. \n∂z/∂t =−ν∂4z/∂x4 −∂2z/∂x2\t −z∂z/∂x, t≥0 \t\t \nz (0) = z0 (x) , \n \nwhere the instability parameter ν > 0. The KS equation is a nonlinear partial differential equation that is first-order in time and fourth-order in space. It models reaction-diffusion systems and is related to various pattern formation phenomena where turbulence or chaos appear. For instance, it models long wave motions of a liquid film over a vertical plane. When the instability parameter ν < 1, this equation becomes unstable. This is shown by analyzing the stability of the linearized system and showing that the nonlinear C0- semigroup corresponding to the nonlinear KS equation is Fr ́echet differentiable. \n \nThere are a number of papers establishing the stabilization of this equation via boundary control. In this thesis, we consider distributed control with a single bounded feedback control for the KS equation with periodic boundary conditions. First, it is shown that sta- bilizing the linearized KS equation implies local asymptotical stability of the nonlinear KS equation. This is done by establishing Fr ́echet differentiability of the associated nonlinear C0-semigroup and showing that it is equal to the linear C0-semigroup generated by the linearization of the equation. Next, a single state-feedback control that locally asymptot- ically stabilizes the KS equation is constructed. The same approach to stabilize the KS equation from one equilibrium point to another is used. \n \nFinally, the solution of the uncontrolled/state-feedback controlled KS equation is ap- proximated numerically. This is done using the Galerkin projection method to approximate infinite-dimensional systems. The numerical simulations indicate that the proposed Lyapunov-indirect method works in stabilizing the KS equation to a desired state. Moreover, the same approach can be used to stabilize the KS equation from one constant equilibrium state to another.
Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.
Comment cette classification a été obtenuedéplier
Prédiction machine sur la base complète
Imitation des enseignantsNi prévalence calibrée, ni vérité terrain. Validation humaine à venir. Le volet Gemma est une étiquette directe du modèle pour chaque travail de la base, lue sur la notice réduite au titre. Le volet Codex est un classifieur appris des 10 348 étiquettes directes de Codex et calibré sur les taux pondérés de l'échantillon; les champs sans appui suffisant ne portent aucun appel Codex. Le mode candidate est l'union des deux volets; le consensus est leur intersection. Ces sorties portent le statut machine_predicted_unvalidated et ne sont pas des étiquettes humaines.
Scores du classifieur distillé par catégorie (deux têtes)
| Catégorie | Codex | Gemma |
|---|---|---|
| Métarecherche | 0,000 | 0,001 |
| Méta-épidémiologie (sens strict) | 0,001 | 0,000 |
| Méta-épidémiologie (sens large) | 0,001 | 0,000 |
| Bibliométrie | 0,000 | 0,000 |
| Études des sciences et des technologies | 0,000 | 0,001 |
| Communication savante | 0,001 | 0,001 |
| Science ouverte | 0,001 | 0,001 |
| Intégrité de la recherche | 0,001 | 0,001 |
| Charge utile insuffisante (le modèle a refusé de juger) | 0,002 | 0,000 |
Scores machine (provisoires)
Les deux têtes enseignantes du modèle étudiant, lues sur ce travail. Un score ordonne la base pour la relecture; il n'affirme jamais une catégorie, et le statut de validation accompagne chaque rangée tel quel.
Scores de référence d'un modèle non mature (critères de maturité non atteints, 7 itérations). Un score ordonne; il n'affirme jamais une catégorie.
score_only:v0-immature-baseline · tel quel depuis la passe de notation : score_only signifie que le nombre peut ordonner les travaux, et qu'aucune étiquette de catégorie n'en découleClassification
machine, non validéePrédiction automatique; un appel candidat d’une seule source (Gemma direct ou Codex distillé), pas un consensus.
Le détail, modèle par modèle et score par score, se trouve en fin de page sous « Comment cette classification a été obtenue ».