Abstract
In chapters 8 and 9 the essentials of the input-output approach to control and regulation have been outlined in a large class of systems of physical and biological interest. It has already been mentioned that the mathematical tools used in developing that approach are of quite a different nature from those employed earlier; instead of variational principles and their Euler equations, the Laplace transform and linear nth order differential equations were found. If the theory had been extended further, such things as the Cauchy integral theorem, and other aspects of the theory of complex variables would have been seen.
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Notes to Chapter 10
Much important and modern pure mathematics traces its roots to the study of dynamical systems in physics. Two of the most seminal books in modern mathematics, namely Poincare’s Methodes Nouvelles de Méchanique Celeste (reprinted) Dover, New York 1957 and G. D. Birkhoff ‘Dynamical Systems’, Am. math. Soc. Colloquium Publ., Vol. 9 (1927), pioneered. From these works there developed the recent ‘global’ approaches to the theory of differential equations (see Note 2, Chapter 6, for references), much of importance in differential geometry and the theory of continuous groups (see W. H. Gottschalk and G. A. Hedlund, Topological Dynamics’, Am. math. Soc. Colloquium Publ, Vol. 36, 1955), the ‘global’ approach to variational problems which is now called Morse theory (see Marston Morse, ‘The Calculus of Variations in the Large’, Am.math. Soc. ColloquiumPubl.,Vol. 18,1934),and most of ergodic theory and the theory of measure-preserving transformations (for example P. R. Halmos, ‘Lectures on Ergodic Theory’, Math. Soc, Japan, 1956). All these studies are now self-contained mathematical disciplines, and the theory of dynamical systems continues to pose problems whose solutions will doubtless lead to the development of comparable new disciplines in the future,
The subsequent discussion is based largely on papers of R. E. Kaiman, who has analysed linear dynamical systems in some detail, The interested reader should consult the following papers : R. E. Kaiman, ‘Contributions to the Theory of Optimal Control’, Bol. de la Soc. Mat. Mexicana (1960), 102–119; R. E. Kaiman, ‘The Theory of Optimal Control and the Calculus of Variations’, RIAS Tech. Rep. 61–3; R. E. Kaiman, ‘Mathematical Description of Linear Dynamical Systems’, /. SIAM Control Ser. A, 1 (1963) 152–91; R. E. Kaiman, Y. C. Ho and K. S. Narendra, ‘Controllability of Linear Dynamical Systems’, Contributions to Differential Equations 1 (1964) 189–212.
This definition of output is somewhat more general than in the older theory. Kalman’s analysis of linear systems (see preceding note for references) shows that there may exist observables of a dynamical system which are not controllable; i.e. which do not depend exclusively on the nature of the inputs to the system. Roughly, this happens when the system possesses degrees of freedom which are not amenable to direct external observation.
See especially the paper of Kaiman, Ho and Narendra, and the J. SIAM paper op. cit.
See for example L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York, 1962.
R. Bellman, Dynamic Programming, Princeton, 1957; R. Bellman, Adaptive Control Processes, Princeton University Press, Princeton, 1961.
An interesting paper in this respect is S. E. Dreyfus, ‘Dynamical Programming and the Calculus of Variations’, J. Math. Anal. Appl., 1 (1960) 228–39. This paper should be compared with the RIAS Report of Kaiman op. cit. for a feeling of the interrelation between these ideas.
The reader is referred to the recent work by Brian C Goodwin, Temporal Organization in Cells: A Dynamic Theory of Cellular Control Processes, Academic Press, New York, 1963; Goodwin’s treatment represents at least one direction in which the ideas sketched in this chapter will provide a unifying influence in theoretical biology. The point of departure of Goodwin’s work is the description of the various kinds of chemical control mechanisms in cells discussed in Section 9.5 above, mainly in the mathematical terms which have been developed in Chapter 8.
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Rosen, R. (1967). Dynamical Systems and Control Theory. In: Optimality Principles in Biology. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-6419-9_10
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DOI: https://doi.org/10.1007/978-1-4899-6419-9_10
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