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SUMMARY:Stability results of dissipative systems via the frequency domain
approach
DTSTART;VALUE=DATE-TIME:20160617T073000Z
DTEND;VALUE=DATE-TIME:20160617T081500Z
DTSTAMP;VALUE=DATE-TIME:20200716T045155Z
UID:indico-contribution-2966@indico.math.cnrs.fr
DESCRIPTION:Speakers: Serge Nicaise (Laboratoire de Mathématiques et de l
eurs Applications de Valenciennes)\nThe frequency domain approach goes bac
k to J. Prüss [Trans. Amer. Math. Soc. 284 (1984)\, 847-857] and F. L. Hu
ang [Ann. Differential Equations 1 (1985)\, 43-56] that show that a $C_0$
semigroup $(e^{tA})_{t\\geq 0}$ of contractions in a Hilbert space $H$ is
exponentially stable if and only if the resolvent of $A$ is uniformly boun
ded on the imaginary axis. Afterwards Z. Liu and B. Rao [Z. Angew. Math.
Phys. 56 (2005)\, 630-644]\, C. J. K. Batty and T. Duyckaerts [J. Evol.
Equ. 8 (2008)\, 765-780]\, and\nA. Bátkai\, K.-J. Engel\, J. Prüss and
R. Schnaubelt [Math. Nachr. 279 (2006)\, 1425-1440] have given some suffi
cient conditions on the behavior of the resolvent of $A$ on the imaginary
axis that guarantee an almost polynomial decay of the semigroup. Finally a
n optimal result about the polynomial decay was found by A. A. Borichev an
d Yu. V. Tomilov [Math. Ann. 347 (2010)\, 455-478]. This approach is a pow
erful tool for the study of the decay of the semigroup associated with co
ncrete dissipative systems since it reduces to the study of the resolvent
on the imaginary axis. \n\nIn our talk\, we will first recall these two re
sults and then illustrate them on two particular dissipative systems\, nam
ely a generalized telegraph equation [Z. Angew. Math. Phys. 66 (2015)\, 3
221-3247] and a dispersive medium model (joint work with C. Scheid (Univ.
Nice)).\n\nhttps://indico.math.cnrs.fr/event/939/contributions/2966/
LOCATION: Salle de Réunion - Bâtiment M2
URL:https://indico.math.cnrs.fr/event/939/contributions/2966/
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