On the von Karman Equations - Initial-Boundary Value Problems and Stabilization
Authors
Pedersen, Bjarne ; Christensen, Henrik Vie
Term
4. term
Education
Publication year
2002
Abstract
Denne afhandling undersøger von Karman-ligningerne i både stationær (tidsuafhængig) og tidsafhængig form. Første del introducerer de nødvendige forudsætninger: et produkt i Sobolev-rum (funktionsrum til analyse af differentialligninger), en sætning om eksistens, entydighed og regularitet for elliptiske randværdiproblemer, nogle resultater om den biharmoniske operator og Monge–Ampère-formen samt grundlæggende teori om dynamiske systemer og stabilitet. I anden del behandles selve von Karman-ligningerne: Først vises det, at en svag løsning (et generaliseret løsningsbegreb egnet til PDE-analyse) er kontinuert i tid. Derefter bevises eksistens og entydighed for de tidsafhængige von Karman-ligninger. Dernæst gives et eksistens- og regularitetsresultat for de stationære von Karman-ligninger. Afslutningsvis præsenteres et stabiliseringsresultat for de tidsafhængige ligninger med rand-feedback (feedback anvendt på randen for at styre systemet mod en stabil tilstand).
This thesis studies the von Karman equations in both stationary (time-independent) and time-dependent forms. The first part introduces the needed background: a product on Sobolev spaces (function spaces used to analyze differential equations), existence, uniqueness, and regularity results for elliptic boundary value problems, results on the biharmonic operator and the Monge–Ampère form, and basic theory of dynamical systems and stability. The second part focuses on the von Karman equations themselves: it first shows that a weak solution (a generalized notion of solution used in PDE analysis) is continuous in time. It then proves existence and uniqueness for the time-dependent von Karman equations, followed by existence and regularity for the stationary equations. The thesis concludes with a stabilization result for the time-dependent equations using boundary feedback (feedback applied at the boundary to drive the system toward a stable state).
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