Global Attractors of Non-Autonomous Dynamical and Control Systems: 2nd Edition

616 Pages·2015·4.188 MB·English

by  David N Cheban

Global Attractors of
Non-Autonomous Dynamical
and Control Systems
2nd Edition
9297hc_9789814619820_tp.indd 1 20/11/14 2:35 pm
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Interdisciplinary Mathematical Sciences – Vol. 18
y
0 x
Global Attractors of
Non-Autonomous Dynamical
and Control Systems
2nd Edition
David N Cheban
State University of Moldova, Moldova
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Library of Congress Cataloging-in-Publication Data
Cheban, David N.
Global attractors of non-autonomous dynamical and control systems / by David N Cheban (State University
of Moldova, Moldova). -- 2nd edition.
pages cm. -- (Interdisciplinary mathematical series, ISSN 1793-1355 ; vol. 18)
ISBN 978-9814619820
1. Attractors (Mathematics) 2. Differentiable dynamical systems. 3. Differential equations. I. Title.
QA614.813.C43 2014
514'.74--dc23
2014037408
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A catalogue record for this book is available from the British Library.
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November20,2014 13:34 GlobalAttractorsofNon-Autonomous...-9.75inx6.5in b1979-fm pagev
Dedicated to my wife Ivanna
and my children Olga and Anatoli
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July25,2013 17:28 WSPC-ProceedingsTrimSize:9.75inx6.5in icmp12-master
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Preface to the Second Edition
Inthesecondeditionofthisbook,wehaveaddedSection11(Chapter1)andChap-
ters 15–17. The new chapters include some results concerning Control Dynamical
Systems (the global attractors, asymptotic stability of switched systems, absolute
asymptoticstabilityofdifferential/differenceequationsandinclusions)publishedin
the works of the author in the last years.
In Section 11 of Chapter 1, we introduce the notion of Birkhoff’s center for
compact dissipative dynamical systems. We establish the analogous of Birkhoff’s
theoremforone-sideddynamicalsystems(bothwithcontinuousanddiscretetimes)
with non-compact space having a compact global attractor. We also establish the
relation between Levinson center, chain recurrent set and center of Birkhoff for
compact dissipative dynamical systems.
Chapter 15 is dedicated to the study of the problem of existence of compact
global attractors for control systems (both with continuous and discrete time) and
to the description of its structure.
The aim of Chapter 16 is to study the problem of uniform asymptotic stability
of the switched system
x(cid:1) =fν(t)(x) (x∈En), (0.1)
where ν :R+ (cid:2)→{1,2,...,m} is an arbitrary piecewise constant function, En is an
n-dimensional Euclidien space, and R+ :=[0,+∞).
In this chapter, we also study the problem of uniform asymptotic stability of
the discrete switched system
u(t+1)=fν(t)(u(t)) (u∈En), (0.2)
where ν :Z+ (cid:2)→P :={1,2,...,m} is an arbitrary piecewise constant function, En
is an n-dimensional Euclidien space, and Z+ :={0,1,2,...}.
Chapter 17 is dedicated to the study of absolute asymptotic stability of differ-
ential/difference equations and inclusions. We establishthe relationbetween linear
inclusions and non-autonomousdynamicalsystems (cocycles). In the frameworkof
general non-autonomous dynamical systems (both linear and nonlinear), we study
the problem of asymptotic stability and absolute asymptotic stability for discrete
vii
November20,2014 13:34 GlobalAttractorsofNon-Autonomous...-9.75inx6.5in b1979-fm pageviii
viii Global attractors of non-autonomous dynamical and control systems
linear inclusions. We also study asymptotic stability of switched systems. We
show that every switched system generates a non-autonomous dynamical system
(cocycle). Using this fact we apply the ideas and methods of the theory of non-
autonomous dynamical systems to the study the problem of asymptotic stability
of different classes ofswitched systems (linear systems, homogeneoussystems,slow
switched systems etc).
The results given in Section 11 (Chapter 1) and Chapters 15–17 of this book
belong to the author. The results of Sections 16.6–16.9 are obtained jointly with
Boularas [(38)] and Sections 17.4–17.7 jointly with Mammana [(130)]–[(133)]. The
basic results of this book are contained in the lectures which author has given for
many years for the students of State University of Moldova.
Acknowledgment: The research described in this publication, in part, was
possible due to grant FP7-PEOPLE-2012-IRSES-316338.
Chi¸sin˘au, July 2014
D. N. Cheban
[email protected], [email protected]
http://www.usm.md/davidcheban
November20,2014 13:34 GlobalAttractorsofNon-Autonomous...-9.75inx6.5in b1979-fm pageix
Preface to the First Edition
In the qualitative theory of differential equations, non-local problems play an im-
portantrole,especiallywithregardtoquestionsofboundedness,periodicity,almost
periodicity, Poissonstability, asymptotic behavior, dissipativity, etc.
The present work takes a similar approach and is dedicated to the study of
abstract non-autonomous dissipative dynamical systems and their application to
differential equations.
In applications, there often occur systems
(cid:1)
u =f(t,u), (0.3)
which have every one of their solutions driven into fixed bounded domain and kept
there under further increase of time, because of natural dissipation. Such systems
are called dissipative ones [(169)]–[(171)],[(323)]–[(325)],[(383)], [(384)]. Solutions
of dissipative systems are called limit (finally) bounded [(383, 384)].
Dynamical systems occur in hydrodynamics when studying turbulent phenom-
ena, meteorology, oceanography, theory of oscillations, biology, radio engineering
and other domains of sciences and engineering techniques related to the study of
asymptotic behavior. Lately, interest in dissipative systems has increased even
more because of intensive elaboration of strange attractors (see, e.g., [(181, 287,
350, 360)]).
There are plenty of dedicated works in the study of the dissipative systems,
beginning from the classical works of N. Levinson. Among works on dissipative
systems of ordinary differential equations, two directions can be made out.
To the first belong works which contain some conditions assuring the dissipa-
tivity of system (0.3), some class or concrete system, representing theoretical or
applied interest. Examples are works of Atrashenok [(10)], Demidovich [(169)]–
[(171)], Zubov [(394)]–[(396)], Matrosov [(300)], Nemytski [(311)]–[(312)], Pliss
[(323)],Schennikov[(352,353)],Corduneanu[(158)],Levinson[(283)],Pavel[(317)]–
[(319)], Reissig [(325)], Talpalaru [(363)] and a lot of other authors.
To the second direction belong works in which inner conditions of dissipative
systemsarestudied,thatis,conditionsrelatingtothecharacterofbehaviorofsolu-
tionofthesystemwhenassumingitsdissipativity,fordifferentclassesofdifferential
ix