Issue |
ESAIM: COCV
Volume 9, February 2003
|
|
---|---|---|
Page(s) | 247 - 273 | |
DOI | https://doi.org/10.1051/cocv:2003012 | |
Published online | 15 September 2003 |
How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance
1
Department of Electrical and Electronic Engineering,
Imperial College London, Exhibition Road, London SW7 2BT, UK; G.Weiss@imperial.ac.uk.
2
Department of Mathematics, University of Nancy I, BP. 239,
54506 Vandœuvre-les-Nancy, France; Marius.Tucsnak@iecn.u-nancy.fr.
Received:
17
January
2003
Let A0 be a possibly unbounded positive
operator on the Hilbert space H, which is boundedly invertible. Let
C0 be a bounded operator from to another Hilbert
space U. We prove that the system of equations
determines a well-posed linear system with input u and output y.
The state of this system is
where X is the state space. Moreover, we have the energy identity
We show that the system described above is isomorphic to its dual, so
that a similar energy identity holds also for the dual system and
hence, the system is conservative. We derive various other properties
of such systems and we give a relevant example: a wave equation on a
bounded n-dimensional domain with boundary control and boundary
observation on part of the boundary.
Mathematics Subject Classification: 93C25 / 93C20 / 35B37
Key words: Well-posed linear system / operator semigroup / dual system / energy balance equation / conservative system / wave equation.
© EDP Sciences, SMAI, 2003
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