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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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