UK Nonlinear News <http:../../>, March 2003 <../>

 

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The Dynamics of Control

 

 

Fritz Colonius and Wolfgang Kliemann

 

 

Reviewed by Jaroslav Stark

 

Birkhauser, 2000, 629 pages

Hardback, 72.00 (Amazon)

ISBN: 0-8176-3683-8

 

Dynamical systems and control theory are two subjects that ought to have

a great deal in common. Yet, somehow, the two communities seem to have

evolved largely independently, and often speak somewhat different

languages. As a result, there is far less interaction between the two

fields than there ought to be. In my opinion, this is a great loss to

both sides, but despite the efforts of various committed individuals,

the barriers between the two areas are proving remarkably difficult to

break down. The key issue is perhaps that in control theory, one has to

consider arbitrary time-varying inputs to a system, whilst nonlinear

dynamics has concentrated on either autonomous systems, or periodically

driven ones (which can be treated as autonomous through the usual device

of regarding time as an extra state space variable).

 

The present volume is an impressive, and intriguing, attempt to develop

an integrated account of the mathematical connections between nonlinear

control, dynamical systems and time-varying perturbed systems. The key

unifying concept is to regard all the possible time-varying inputs as a

shift space (usually infinite dimensional) driving the dynamical system

of interest. In this way a driven system becomes a so called /skew

product/ over the driving shift space and can now be regarded as an

autonomous system. In essence, this is the same principle as that used

to convert periodically driven systems into autonomous ones, except that

instead of enlarging the system by a single variable, we typically add

an infinite number. Nevertheless, this device allows the use of many

standard techniques and results from nonlinear dynamics. Such an idea is

also central to some modern approaches to stochastic dynamics, and in

particular to Ludwig Arnold?s seminal work on /Random Dynamical Systems/

(reviewed in UK Nonlinear News 17, August 1999). This commonality is no

coincidence, since Arnold was Wolfgang Kliemann?s PhD supervisor, and

Fritz Colonius also studied at Bremen. In some ways the present volume

can be seen as a continuation of Arnold?s programme, applied

specifically to control systems. The main difference in the present

volume is that more regularity is assumed, so that the resulting skew

product is continuous. This allows the authors to apply a variety of

concepts from topological dynamics and ergodic theory. The exciting

aspect of this is that there turn out to be intimate connections between

control theory properties of the control system, and the dynamical

properties of the skew product. Thus for instance there is a one-one

correspondence between control sets and topologically mixing invariant

sets for the skew product.

 

The end result is a treatment which is highly accessible to someone

familiar with the basic concepts of dynamical systems. At the same time,

the particular applications will be largely unknown, and hence the book

provides a wealth of new ideas to explore. I would therefore recommend

almost anyone with a background in nonlinear dynamics to read it for

this reason alone. Having no expertise in control theory, I am unable to

judge how this volume will be perceived within that community, but again

would imagine that it could provide a fresh perspective for those

interested in the more theoretical aspects of control.

 

/ /

 

/ UK Nonlinear News / would like to thank Birkhauser for providing a

review copy of this book.

 

A listing of books reviewed in UK Nonlinear News is available

<../../uknonl-books.html>.

 

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