On a unified geometric theory of control. (Q2782032)

The monograph under review is unique in many ways. The reviewer was fortunate to receive preprints of articles of Professor Boltyanskiĭ and also of Butkovskiĭ in collaboration with the coauthors of this work. Professor \textit{V. G. Boltyanskiĭ} is one of the great control theorists, coauthor with \textit{L. S. Pontryagin, R. V. Gamkrelidze} and \textit{E. F. Mishchenko} of the classic monograph ``Mathematical Theory of Optimal Processes'' [Nauka, Moscow (1961); English translation Interscience, New York (1962; Zbl 0102.32001)], which revolutionized control theory around 1960. The authors claim that this book is primarily about control theory. Perhaps it is, but in the reviewer's opinion it is much more than that. It is a philosophical look at all of mathematics, perhaps all sciences. Indeed it contains some new and important developments in control theory pioneered by Butkovskiĭ in a sequence of papers in which he advanced a unified geometric control theory. See for example (and this is one of several papers on this topic) Geometrical Structures in Control Theory by \textit{A. G. Butkovskiĭ} [Syst. Sci. 25, No. 1, 121--126 (1999; Zbl 1068.93506)], where he introduces a metric space in which the metric is not symmetric \((\rho (a,b)\neq\rho (b,a))\), nor is it true that \(\rho(a,b) >0\) when \(a\neq b\). He also introduces the concept of a metric chain. A secondary metric is also introduced, corresponding to a weaker topology. The optimality of the cost functional corresponds to the shortest distance in this metric space. But all this and much more that strictly concerns control theory, as it is identified at this time, is largely relegated to the chapters 5, 6, and 7. In the remainder of this work, the authors take the viewpoint that almost any mathematical problem can be restated in the jargon of control theory.NEWLINENEWLINENEWLINEIn particular, the elementary concepts of control theory: structure of a system, abstract space describing its geometric interpretation, and other mathematical ideas, including a fundamental concept of symmetry, are common to diverse areas of mathematics, including control theory and cybernetics. The authors tie the concept of control not only to modern physics, but to all sciences, to logic, to cybernetics, to Markov processes, to the finding of relations between civilization and thinking processes, human languages and formal languages.NEWLINENEWLINENEWLINEAn almost poetic sub-chapter discusses Civilization and Culture. It contains a discussion of the concept of God as interpreted in our culture. It ends in the conclusion that the concept of Deity is that of an unimaginably complex control system. However, science is given by a quotation, ``It is the manner in which God informs us how He created (or rather designed) our universe.''NEWLINENEWLINENEWLINEThis is followed by the \(100\%\) effectiveness of mathematics principle. In a largely experimental science, say medicine, experiments produce data. It is the work of theoreticians to find some meaning in such collected data. Then some theory is announced which hopefully does not contain self-contradictory consequences. The most exact way to attain such a theory and check its consequences is given by the application of mathematics. And it should be mathematics which uses a universal language, creates models of reality, and supplies algorithms for relevant computations. The \(100\%\) effectiveness of mathematics (explained in the text) helped to make discoveries of relativity and quantum mechanics, which in turn have chosen mathematical concepts that best approximated the relativistic or quantum ``reality''. Some scientists described mathematics as a science of abstract structures and symmetries. Famous men, among them Wigner and von Neumann, discussed the unexplained effectiveness of pure mathematics. ``Applied mathematics'' combines its ``pure'' aspects with efforts towards a morphism between mathematics and reality. An important concept is that of fibering a space such that, for example, a function synthesizing a control can be regarded as a cross-section of fibers. A point \(x\) in the base of fibers \(X\) may be associated with a fiber \(U_x\) in the set of all fibers \(U\). The ideas, related to the algebraic topology formalism, appeared earlier in the writings of A. G. Butkovskiĭ.NEWLINENEWLINENEWLINERoughly 140 pages contain a short introduction to sometimes original axiomatic foundations of all mathematics: logic, topology, analysis, and of course control theory. The authors' creation of Huygens' metric is discussed in several chapters, stressing its importance to control theory of systems with distributed parameters.NEWLINENEWLINENEWLINEThere are other novel approaches to control theory. For example there is a very interesting discussion of the relation between arbitrary control systems and Zermello's axiom of choice. Set-theoretic topological spaces, algebraic spaces and category theory are all given a quick insight from an advanced point of view, and are shown to fit into the geometric theory of control. The authors point out how category theory was shown to fit quantum mechanical field theory by M. Atiyah. Using the fibering of spaces together with category theory, they propose to unify the different aspects of the present state of control theory. The reviewer comments that this is a departure from accepted ideas of input, output and a state function. The control system, described here by its behavior, which is a collection of trajectories (or fibers), has been considered recently by other researchers. S. Shankar and Jan Willems offered a similar argument concerning the use of homological ideas in the control of systems with distributed parameters (PDEs). See the reference \textit{S. Shankar} and \textit{J. C. Willems} [Behavior of \(n\)-D systems, Second international workshop for \(n\)-D systems, June 2000, Czecha Castle, Poland, 23--30 (2000; Zbl 0971.93016)].NEWLINENEWLINENEWLINEThe authors also point out that Feynman diagrams fit into this scheme when regarded as categorical diagrams, which are frequently used in category theory. This approach also fits naturally into the programming of computers. They suggest that a monstrous size program of unification of all these areas is in some sense analogous (``isomorphic''?) to the proposed idea of the unique geometric theory of control. They offer an example of replacement of the usual block diagram of control theory by a purely algebraic structure.NEWLINENEWLINENEWLINEChapter 5 introduces the concepts of ordering, lattices, order preserving decompositions, assigning a metric, or a generalized metric, and the concept of generalized boundary of a set \(A\), denoted by \(A/\partial\), discussed in detail. All this fits into the geometric theory of control. Chapter 6 discusses the theory of differential inclusions with applications to control theory. The Lagrangian for such a system was first introduced by Minkowski, who called it the calibration function. It is the starting point in a new derivation of Pontryagin's maximality principle for such systems. The authors offer five (new) theorems which, in the reviewer's opinion, will be crucial to a further development of these ideas. The final Chapter 7 contains some of the recent ideas on the extensions of Pontryagin's maximality principle. The last theorem ending the text of this book relates existence of a mimmum for an analog of Pontryagin's Hamiltonian to the existence of a closed differential form. NEWLINENEWLINENEWLINEThis is an amazing monograph that, in the reviewer's opinion, should be translated into English (and perhaps other languages). The authors feel that perhaps we are on the verge of another revolutionary step, similar to the introduction of functional analysis into the approaches mainly based on ideas of the revolution brought about by Newton and Leibniz.