UK Nonlinear News , November 1998

# Partial Stability and Control

## By V. I. Vorotnikov

### Reviewed by Stuart Townley (University of Exeter)

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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