The synthesis of dynamical systems

Author:
R. W. Brockett

Journal:
Quart. Appl. Math. **30** (1972), 41-50

DOI:
https://doi.org/10.1090/qam/99741

MathSciNet review:
QAM99741

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Abstract | References | Additional Information

Abstract: A significant part of contemporary applied mathematics is concerned directly with communication, control and computation. In these fields many of the central problems involve the *synthesis* of algorithms, or dynamical systems, as opposed to the *analysis* of dynamical systems which predominates in mathematical physics. Arithmetic and numerical algorithms, finite-state machines and electrical filters are examples of the types of dynamical systems which are frequently needed to operate on data, in continuous or discrete form, and to produce data on a compatible time scale. In this paper we discuss the scope and success of some of the synthesis procedures currently available to treat these problems.

**[1]**R. Abraham,*Foundations of mechanics*, W. A. Benjamin, Inc., N. Y., 1967**[2]**George C. Newton Jr., Leonard A. Gould, and James F. Kaiser,*Analytical design of linear feedback controls*, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London;, 1957. MR**0089796****[3]**O. Heaviside,*Electrical Papers*, Vol. I and II, Macmillan and Co. 1892 (see Vol. II, pp. 353-374)**[4]**Lord Rayleigh,*The reaction upon the driving-point of a system executing forced harmonic oscillations of various periods, with applications to electricity*, Phil. Mag.**21**, 369-381 (1886);*Scientific papers*, Vol. 2, pp. 475-485**[5]**D. E. Knuth,*The Art of Computer Programming*, Vol. 2, Addison-Wesley, 1969**[6]**A. M. Turing,*On Computable Numbers, with an Application to the Entscheidungsproblem*, Proc. London Math. Soc. (2)**42**(1936), no. 3, 230–265. MR**1577030**, https://doi.org/10.1112/plms/s2-42.1.230**[7]**R. E. Kalman, P. L. Falb, and M. A. Arbib,*Topics in mathematical system theory*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1969. MR**0255260****[8]**R. W. Brockett,*Finite Dimensional Linear Systems*, John Wiley and Sons, N. Y., 1970**[9]**A. Nerode,*Linear automaton transformations*, Proc. Amer. Math. Soc.**9**(1958), 541–544. MR**0135681**, https://doi.org/10.1090/S0002-9939-1958-0135681-9**[10]**R. W. Brockett and A. Willisky,*Finite group-homomorphic sequential machines*, IEEE Trans. on Automatic Control (to appear)**[11]**R. W. Brockett and Jacques Willems,*Least squares optimization for stationary linear partial difference equations*, in*Proc. IFAC Symposium on The Control of Distributed Parameter Systems*, Banff, Canada, 1971**[12]**C. M. Fiduccia,*Fast matrix multiplication*, in*Proc. Third Annual ACM Symposium on Theory of Computing*, Shaker Heights, Ohio, May 3-5, 1971**[13]**A. M. Ostrowski,*Solutions of equations and systems of equations*, Academic Press, N. Y., 1960**[14]**J. F. Traub,*Optimal iterative processes: theorems and conjectures*, Information processing 71 (Proc. IFIP Congr., Ljubljana, 1971) North-Holland, Amsterdam, 1972, pp. 1273–1277. MR**0458852****[15]**J. C. Butcher,*On the convergence of numerical solutions to ordinary differential equations*, Math. Comp.**20**(1966), 1–10. MR**0189251**, https://doi.org/10.1090/S0025-5718-1966-0189251-X**[16]**David S. Evans,*Finite-dimensional realizations of discrete-time weighting patterns*, SIAM J. Appl. Math.**22**(1972), 45–67. MR**0378915**, https://doi.org/10.1137/0122006**[17]**D. S. Evans,*Generalized linear multistep methods: a weighting pattern approach to numerical integration*, Ph.D. Thesis, M.I.T. 1969**[18]**E. Picard,*Memoire sur la théorie des equations aux derivées partielles et la méthode des approximations successives*, J. Math. Pures Appl. (**5**)**6**, 423-441 (II 1) (1890)**[19]**Jan Willems,*The analysis of feedback systems*, M.I.T. Press, Research Monograph No. 62, 1971**[20]**I. J. Gorille,*On the application of discrete time stability criteria to numerical analysis*, M.S. Thesis, Dept. of Electrical Engineering, M.I.T., 1966**[21]**Edward Nelson,*Dynamical theories of Brownian motion*, Princeton University Press, Princeton, N.J., 1967. MR**0214150**

Additional Information

DOI:
https://doi.org/10.1090/qam/99741

Article copyright:
© Copyright 1972
American Mathematical Society