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Well-posed linear systems. (English) Zbl 1057.93001
Encyclopedia of Mathematics and Its Applications 103. Cambridge: Cambridge University Press (ISBN 0-521-82584-9/hbk). xviii, 776 p. (2005).
This book is a treatise on realisation theory for abstract linear systems where a central issue is the notion of well-posedness. The views of a system in the time-domain as an input/output map, in the frequency domain via the transfer function or in the state space form (and the associated system node), are related. Key properties are causality, time-invariance, concatenability of solutions.
An introduction and overview of 27 pages allows the reader to get a survey of the topics treated.
Chapter two studies well-posed input/output systems arising naturally from a system node. Well-posedness is defined for input, state and output spaces taken as Banach spaces, the input and output signals belong to $$L_p$$ spaces defined on the corresponding spaces, and usual existence of the state and continuous dependence on the initial state and on the control are involved. $$1\leq p\leq\infty$$ and the case $$p=\infty$$ is somewhat special but the specificity is reduced when one limits the focus to regulated functions. The author studies then boundedness properties for such systems.
Chapter three presents the basic theory of $$C_0$$ semigroups (generator, contraction semigroups, dual semigroups (reflexive and nonreflexive cases), resolvent, Cauchy problem, symbolic calculus, analytic semigroups, growth properties, Laplace transform, invariant subspaces).
Chapter four studies the realization theory from a system node. The important issue here is to find the right spaces on which the operators derived from the node are acting. Finding the feedthrough operator is obtained only in simple cases (bounded control or observation operators). Realisation theory in more general situations (not necessarily “$$L_p$$ or regulated well-posed”) are considered and some results obtained. Specific cases are examined (diagonal system, normal system, delay line, Lax-Phillips semigroup).
Chapter five attempts to find the feedthrough operator in the context of the previous chapter. The central notion of compatible system is introduced. One gets a feedthrough operator provided the observation operator is extended to a certain nonunique space. The set of compatible systems does not differ much from “$$L_p$$ or regulated well-posed” systems. The author studies regular systems, i.e., systems whose observation operator can be extended and the result is compatible in the sense just explained. Many examples of regular systems are obtained too. The standard well-posed systems of the book will be regular unless both observation and control operators are “maximally unbounded”.
The rest of the book examines standard systems science concepts in the above context. Time and flow inversion are examined. Anti-causal and dual systems are introduced. Chapter seven is devoted to feedback (and the notion of feedback regularity is examined); duality of a closed-loop system is studied. The next chapter follows in a straightforward way with the notions of stability, stabilizability, coprime factorisation and extensions. Chapter nine focuses on specific issues in realisation theory: minimality, balancing and the key related controllability and observability issues are well integrated in this frame. Chapter ten is devoted to the notion of admissibility, and the control and observation operator are seen to be admissible for several types of semigroups. The role of the Carleson measure theorem is emphasized. Chapter eleven studies passive, conservative and lossless sytems. Sections focus on conservative extensions of conservative systems, conservative realizations, passive realisations. The last chapter is on discrete systems and various associated transformations. The last three chapters restrict the study to Hilbert spaces.
Four appendices provide help on technical matters. A bibliography and an index are included. The chapters end with useful comments and bibliographical information.
This book is well organized and very clearly written. Also the theme is very important and such a monograph was needed. The same editor published already two volumes of I. Lasiecka and R. Triggiani (Zbl 0961.93003 and Zbl 0942.93001) on optimal control of PDEs, and the mathematical level was more advanced, whereby the basics explained here in chapter three are assumed to be known. And these authors plunged into specific physical situations which are claimed to be covered here. Here, moreover, the situation is more general and also more mathematical and it provides an excellent background for the volumes cited above, completing them well. The author informs us in his preface that his main motivation in writing this book was to handle optimal control and impedance passive systems. Both topics are left untouched because of a lack of time and space as well as other constraints. (One other reason is the observation that the theory of optimal control has not the “needed maturity” (p. 8) at the theoretical level; others claim that the theory has been settled in the sixties and the future of the discipline lies in its applications). Let us observe that the initial goals of the author would have led him to consider not well-posed systems (some of them included here making the title not entirely well chosen; this is “the first comprehensive treatment of (possibly non-well-posed) systems” [the preface]), so that he ended up doing other things than those originally envisaged. Let us express our sympathy for this attitude, and let us hope that the author will find the right conditions so that his initial goals can be reached.

##### MSC:
 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory 93C25 Control/observation systems in abstract spaces 93B15 Realizations from input-output data 93B28 Operator-theoretic methods 93B20 Minimal systems representations 93D15 Stabilization of systems by feedback 47D06 One-parameter semigroups and linear evolution equations 47B38 Linear operators on function spaces (general) 47D60 $$C$$-semigroups, regularized semigroups