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Titel:Distributional differential algebraic equations
Autor:Dr. rer. nat. Trenn, Stephan [Autor]
Herausgeber: Universitätsverlag Ilmenau <Ilmenau> [Herausgeber/Verleger]
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Dateien vom 03.08.2009 / geändert 23.09.2009
URL für Lesezeichen:http://www.db-thueringen.de/servlets/DocumentServlet?id=13581
URN (NBN):urn:nbn:de:gbv:ilm1-2009000207
Kollektion:Dissertationen/Habilitationen
Status:Dokument veröffentlicht
Sprache:Englisch
Dokumententyp:Dissertation
Medientyp:Text
Beitragende:Prof. Dr. Ilchmann, Achim [Betreuer/Doktorvater]
Prof. Dr. rer. nat. habil. Babovsky, Hans [Gutachter]
Associate Professor Liberzon, Daniel [Gutachter]
Stichwörter:differential algebraic equations, linear, distributional solutions, switched systems, impulses, regularity
Dewey Decimal Classification:500 Naturwissenschaften und Mathematik » 510 Mathematik » 512 Algebra
Andere Dokumente dieser Kategorie
Evaluationstyp:Für die Langzeitarchivierung vorgesehen
Beschreibungen:Zusammenfassung (dt.)

Abstract:
Linear implicit differential equations of the form Ex'=Ax+f are studied. If the matrix E is not invertible, these equations contain differential as well as algebraic equations. Hence Ex'=Ax+f is called differential algebraic equation (DAE).
A main goal of this dissertation is the consideration of certain distributions (or generalized functions) as solutions and studying time-varying DAEs, whose coefficient matrices have jumps. Therefore, a suitable solution space is derived. This solution space allows to study the important class of switched DAEs.
The space of piecewise-smooth distributions is introduced as the solution space. For this space of distributions, it is possible to define a multiplication, hence DAEs can be studied whose coefficient matrices have also distributional entries. A distributional DAE is an equation of the form Ex'=Ax+f where the matrices E and A contain piecewise-smooth distributions as entries and the solutions x as well as the inhomogeneities f are also piecewise-smooth distributions.
For distributional DAEs, existence and uniqueness of solutions are studied, therefore, the concept of regularity for distributional DAEs is introduced. Necessary and sufficient conditions for existence and uniqueness of solutions are derived. As special cases, the equations x'=Ax+f (distributional ODEs) and Nx'=x+f (pure distributional DAE) are studied and explicit solution formulae are given.
Switched DAEs are distributional DAEs with piecewise constant coefficient matrices. Sufficient conditions are given which ensure that all solutions of a switched DAE are impulse free. Furthermore, it is studied which conditions ensure that arbitrary switching between stable subsystems yield a stable overall system.
Finally, controllability and observability for distributional DAEs are studied. For this, it is accounted for the fact that input signals can contain impulses, hence an "instantaneous" control is theoretically possible. For a DAE of the form Nx'=x+bu, y=cx, with constant, nilpotent N and constant vectors b and c, a normal form is given which allows for a simple characterization of controllability and observability.
Hochschule/Fachbereich:Technische Universität Ilmenau » Fakultät für Mathematik und Naturwissenschaften
Dokument erstellt am: 31.07.2009
Dateien geändert am: 23.09.2009
Promotionsantrag am: 29.05.2009
Datum der Promotion: 28.07.2009