Published by Birkhäuser Verlag AG 1998,

Hardback: 3-7643-3917-9, Price: sFr 158

This monograph gives a detailed and comprehensive
study of the `stability` and `stabilisation` of dynamical
systems with respect to `part of the variables`. The author draws
considerably on his own well-established expertise in the area which he
has gained from
over two-decades of research and demonstrated by numerous publications.

The topic of the monograph has its origins in the classical and
fundamental
works of A.M. Lyapunov, who was the first to formulate the notion of
`partial stability`.
Subsequently, the use of Lyapunov functions became a key tool in the
analysis of the
partial stability problem as instigated by the works of V.V.
Rumyantsev.

The monograph begins with an excellent Introduction. This places the
topic of
the monograph clearly in the historically development of the partial
stability of dynamical systems, includes a detailed account of the
relevant literature and makes use of the monograph's
comprehensive bibliography. Several situations are then described
which help motivate and justify the need for a
study of partial stability/stabilisation.
The introductory chapter is then completed with a review of the
Lyapunov-Rumyantsev Lyapunov function based approach to partial
stability.
This includes detailed definitions of `stability with respect to
part
of the variables` (uniform, asymptotic, global), distinctions between
partial stability and so-called stability of sets, positive definiteness
of (Lyapunov) `V`-functions with respect to part of the variables and
tests for partial stability in terms of
`V`-functions with sign-constant
derivative.
It is worth emphasising that the notion of partial stability,
i.e. stability with respect to part of the variables, is distinct from
extensions of the classical stability (with respect to all the
variables)
to cases when only part of the variables exhibit stability whilst the
remaining variables are bounded. This distinction is made clear in this
monograph. Indeed it is shown that stability with respect to part
of the variables does not even guarantee bounded-continuability of the
remaining variables.

In Chapter 1 the linear problems of stability with respect to part of the variables is introduced. Whilst the case of linear systems is an obvious starting point for developing any general theory, there is added motivation here in that the main technique, that of extending the system by introducing an auxiliary (linear) system, is used subsequently even in the case of nonlinear systems. This extension technique was developed by the author as an alternative to the Lyapunov-function based technique. Based on the auxiliary system, several criteria are developed for testing stability of linear systems with respect to part of the variables. These involve certain rank conditions and provide means for constructing Lyapunov functions via solution of certain matrix equations. Stability with respect to part of the variables is shown to be equivalent to Lyapunov stability with respect to all the variables of the auxiliary system. It is interesting to observe, in the examples, how instabilities occurring in those variables for which partial stability is not obtained cancel out, and that this cancelling ties in with the dependency predicted by the rank tests. The results are generalised to the case of linear systems with non-constant coefficients. Here the main difficulty lies in constructing the auxiliary system. Accordingly various results are obtained depending on the regularity (analyticity, differentiability) of the system data.

In Chapter 2 nonlinear problems of stability with respect to part of the
variables are analysed via the `first approximation`. An intriguing
aspect is that the first approximation of the system is itself
nonlinear. An extension
of this nonlinear first approximation is then determined and it is only
then that linearisation is applied. Stability with respect to all of the
variables of the linearisation of the extended system then implies
stability with
respect to part of the variables for the original system. This approach
is shown to be much more general than results which can be obtained via
first approximations which are linear. The results are applied to absolute
stability with respect to part of the variables for Lur'e systems and to damping
of angular
motions of asymmetric solids with respect to part of the variables.

Chapter 3 considers `essentially' nonlinear problems of stability with respect to part of the variables. A Lyapunov function approach is developed. Indeed, one of the keys tools is the use of (several) vector Lyapunov functionals. Further ideas are developed using so-called ${\bf \mu}$-functions and differential inequalities. The use of differential inequalities is then related to the construction of extended systems.

Chapter 4 uses the results from Chapters 1, 2 and 3 in studying
nonlinear problems of
`stabilisation with respect to part of the variables`. An
interesting
development is
the use of `partial stabilisation` techniques in the stabilisation
of nonlinear systems
`with respect to all of the variables`. This development is
illustrated by applications
to stabilisation of geo-stationary orbits of satellites. The partial
stabilisation
techniques also clarify quite well the effect of reducing the number of
active power
devices used to control such satellites. Further applications to control
of gimballed-gyroscopes, re-orientation of solids and control of systems
(chains) of solids are also considered.

Chapter 5 develops game theoretic problems, again with applications to
the control and re-orientation of solids. Chapter 6 considers extensions of
the results of Chapters 1-4 to the case of `functional-differential
equations`.
Here techniques of `Lyapunov functions` and
`Lyapunov-Krasovskii functionals`
are combined with additional ${\bf \mu}$-functions introduced in Chapter
3.
Finally, Chapter 7 extends the results of Chapters 1-4 to the case of
stochastic systems.

This is a well-written book. At well-over 400 pages there is much to discover and digest. There are numerous useful examples, both of a text-book style, aimed to clarify a definition or detail in a proof, and those of a more significant application-based style. I found the applications to the control of geo-stationary orbit of satellites most interesting and illuminating. Each chapter is clearly introduced and each concludes with an extensive and impressive overview of the literature.

The structure and style reminds me somewhat of the well-known
book `Stability of Motion` by W. Hahn. It remains to be seen whether
this monograph will achieve the same status as this classic study of the
stability theory for dynamical systems. However, I thoroughly enjoyed
reading it and will endeavour to use some of the techniques
in my own research.

*UK Nonlinear News* thanks
Birkhäuser Verlag AG
for providing a copy of this book for
review.

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Last Updated: 23rd October 1998.