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Stabilization of Elastic Systems by Collocated Feedback

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In recent years an extensive literature was devoted to the controllability and
stability of second order infinite dimensional systems coming from elasticity (see,
for instance, Lions [91, 92], Komornik [73], Lasiecka-Triggiani [85], Lasiecka
[81], Slemrod [120], Ammari-Tucsnak [23], Guo and his collaborators [62, 63],
Komornik-Loretti [75], Coron [45], Tucsnak-Weiss [126], and references therein).
According to the classical principle of Russell (see [118]) if a system is uniformly
stabilizable forward and backward in time (plus some technical assumptions) by
using collocated actuators and sensors, then it is exactly controllable by using
the same actuators (i.e., the same input operator). We even refer to [113] for a
very general formulation of this principle. The converse of this assertion is not
true in general but it is true under specific conditions given in [47]. The only
results available in the literature suppose that the input operator is bounded in the
energy space (see Haraux [66]) or they are based on non local feed-backs (see, for
instance, Komornik [74] and the references therein). Applied to PDE systems this
situation leads to non-local feedbacks given in particular by Riccati-type operators.
However for many PDE systems the exponential stability with collocated actuators
and sensors was proved by direct methods using multiplier techniques (see Chen
[39, 40], Lagnese [78], Komornik and Zuazua [76]).
The first aim of this book is to give a class of unbounded input operators for
which exact controllability implies uniform stability by collocated actuators and
sensors. The abstract setting (based on the results from [23]) is presented in Chap. 2
and its validation by concrete dissipative systems is given in Chap. 4.
Our mathematical framework is the following one. Let X be a complex Hilbert
space with norm and inner product denoted, respectively, by jj jjX and .; /X . Let
A be a linear unbounded self-adjoint and strictly positive operator in X. Let D.A1
2 /
be the domain of A1
2 . Denote by .D.A1
2 //0 the dual space of D.A1
2 / with respect
to the pivot space X. Further, let U be a complex Hilbert space (which is identified
with its dual space) with norm and inner product, respectively, denoted by jj jjU
and .; /U and let B 2 L.U; .D.A1
2 //0
/.

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Publisher

London : Springer., 2015

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Language

English

ISBN/ISSN

978-3-319-10900-8

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NONE

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Statement of Responsibility

Kaïs Ammari , Serge Nicaise

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Cataloger

Jemadi

Source

https://link.springer.com/book/10.1007/978-3-319-10900-8

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