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2015 | Buch

Some Backgrounds Kapitel lesen Erstes Kapitel lesen

Stabilization of Elastic Systems by Collocated Feedback

verfasst von: Kaïs Ammari, Serge Nicaise

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

By introducing a new stabilization methodology, this book characterizes the stability of a certain class of systems. The stability (exponential, polynomial, or weaker) for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined with a boundedness property of the transfer function of the associated open loop system. A similar strategy is applied to systems where a delay term is added. The book concludes with many concrete examples. This book is addressed to graduate students in mathematics or engineering and also to researchers with an interest in stabilization and control systems governed by partial differential equations.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Some Backgrounds
Abstract
In the whole book (except in Chaps. 4 and 5), X is a complex and separable Hilbert space with norm and inner product denoted respectively by \(\|\cdot \|_{X}\) and (⋅ , ⋅ ) X .
Kaïs Ammari, Serge Nicaise
Chapter 2. Stabilization of Second Order Evolution Equations by a Class of Unbounded Feedbacks
Abstract
In this chapter we consider second order evolution equations with unbounded feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non uniform decay properties.
Kaïs Ammari, Serge Nicaise
Chapter 3. Stabilization of Second Order Evolution Equations with Unbounded Feedback with Delay
Abstract
We now turn to problems with delays, namely in the same Hilbert setting than in the previous chapter we consider the closed loop system (5): \(\displaystyle{ \left \{\begin{array}{c} x^{{\prime\prime}}(t) + \mathit{Ax}(t) + B_{1}B_{1}^{{\ast}}x^{{\prime}}(t) + B_{2}B_{2}^{{\ast}}x^{{\prime}}(t-\tau ) = 0,\,t > 0 \\ x(0) = x^{0},\,x^{{\prime}}(0) = x^{1}, \\ B_{2}^{{\ast}}x^{{\prime}}(t-\tau ) = f^{0}(t-\tau ),\,0 < t <\tau.\end{array} \right. }\)
Kaïs Ammari, Serge Nicaise
Chapter 4. Asymptotic Behaviour of Concrete Dissipative Systems
Abstract
We study the large time behavior of the solution of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Dirichlet or Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term.
Kaïs Ammari, Serge Nicaise
Chapter 5. Systems with Delay
Abstract
We end up this book by considering different examples of systems with delay for which our abstract framework can be applied.
Kaïs Ammari, Serge Nicaise
Backmatter
Metadaten
Titel
Stabilization of Elastic Systems by Collocated Feedback
verfasst von
Kaïs Ammari
Serge Nicaise
Copyright-Jahr
2015
Electronic ISBN
978-3-319-10900-8
Print ISBN
978-3-319-10899-5
DOI
https://doi.org/10.1007/978-3-319-10900-8

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