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

------------------------------------------------------------------------

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>.

------------------------------------------------------------------------

<< Move to UK Nonlinear News Issue 31 Index Page
<../>.

Page Created: 1 March 2003.

Last Updated: 1 March 2003.

uknonl@amsta.leeds.ac.uk
<mailto:uknonl@amsta.leeds.ac.uk>

------------------------------------------------------------------------