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Partial differential control theory. Vol. 1: Mathematical tools. Vol. 2: Control systems. (English) Zbl 1079.93001

Mathematics and its Applications (Dordrecht) 530. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-7035-X/vol1; 0-7923-7036-8/vol2; 0-7923-7037-6/set). 957 p. (2001).
This very important monograph (in two volumes) presents an algebraic framework for describing linear and nonlinear control systems. The framework considers in a unified way systems described by ordinary differential equations and systems described by partial differential equations. The algebraic approach for dealing with partial differential equations presented here challenges deeply the one relying on functional analysis. Here the author looks for formal definitions, properties and tests.
This first volume introduces the necessary mathematical background. The author writes on page 58: “We present for the first time a coherent parallel between the formal theory of differential operators and the formal theory of differential modules”. And he adds: “We repeat once more that the simplest way for any reader to understand the novelty of this book is to open simultaneously Bourbaki’s treatise, Palamodov’s book and Kashiwara’s thesis and to wonder how these works can interfere with control theory”. Well, most readers will find this quite arduous since appreciating such books and control theory will probably require years of hard reading.
The contents of the book are: commutative algebra (rings, modules, tensor products), homological algebra (sequences, diagrams, resolutions, functors), differential geometry (jet theory, nonlinear systems, linear systems), differential algebra (derivations, differential rings, differential modules, spectral sequences).
The second volume applies the algebraic methods introduced in the first volume to control systems. First, linear systems are studied. The structural properties (controllability, observability, purity) are characterized. Then, input/output properties (matching, invertibility, causality, poles and zeros) are characterized. Several applications involving physical analogies between thermodynamics, elasticity and electromagnetism are clarified with the author’s perspective. Another part considers nonlinear control systems, and the controllability property is dealt with via linearization. Finally, optimal control is tackled and many examples are provided. A bibliography (225 items) is presented at the end. The small index refers to both volumes.
Many results in this book are claimed to be new and the perspective is claimed to be revolutionary. It is certain that both volumes will be key references in control theory and offer many open fields of research. In the reviewer’s opinion this text is one of the most remarkable ones in the history of control theory.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93C20 Control/observation systems governed by partial differential equations
93B25 Algebraic methods
13N05 Modules of differentials
13N10 Commutative rings of differential operators and their modules
93B05 Controllability
93C10 Nonlinear systems in control theory
93B07 Observability
93B29 Differential-geometric methods in systems theory (MSC2000)
93B18 Linearizations
13C12 Torsion modules and ideals in commutative rings
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