Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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
Full-text PDF Free Access

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.

References [Enhancements On Off] (What's this?)

  • [1] R. Abraham, Foundations of mechanics, W. A. Benjamin, Inc., N. Y., 1967
  • [2] G. C. Newton, Jr., L. A. Gould and J. F. Kaiser, Analytical design of linear feedback controls, John Wiley and Sons, Inc., N. Y. 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 application to the entscheidungs-problem, Proc. London Math. Soc. 42, 230-265 (1936-37) MR 1577030
  • [7] R. E. Kalman, P. Falb, and M. Arbib, Topics in Mathematical System Theory, McGraw-Hill, 1969 MR 0255260
  • [8] R. W. Brockett, Finite Dimensional Linear Systems, John Wiley and Sons, N. Y., 1970
  • [9] A. Nerode, Linear automaton transformation, Proc. Amer. Math. Soc. 9, 541-544 (1958) MR 0135681
  • [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, IFIP Congress Booklet TA-1 (1971), 29-32 MR 0458852
  • [15] J. C. Butcher, On the convergence of numerical solutions to ordinary differential equations, Math. Comp. 20, no. 93, (1966) MR 0189251
  • [16] D. S. Evans, Finite-dimensional realizations of discrete-time weighting patterns, SIAM J. App. Math. 22, (1972) MR 0378915
  • [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] E. Nelson, Dynamical theories of Brownian motion, Mathematical Notes, Princeton University Press, 1967 MR 0214150

Additional Information

DOI: https://doi.org/10.1090/qam/99741
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society