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A geometric approach to linear control and estimation

  • T. E. Duncan
Part I: Survey Lectures
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 16)

Keywords

Riccati Equation Canonical Transformation Stochastic Optimization Problem Lagrangian Plane Singular Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    V. I. Arnol'd, Characteristic classes entering in quantization conditions, Funct.Anal.Appl. 1 (1967), 1–13.Google Scholar
  2. 2.
    H. Bass, Quadratic modules over polynomial rings, Contributions to Algebra, (H. Bass, P. Cassidy, J. Kovacic, eds.), 1–23, Academic Press, New York, 1977.Google Scholar
  3. 3.
    R. W. Brockett, Some geometric questions in the theory of linear systems, IEEE Trans.Auto.Control AC-21 (1976), 444–455.Google Scholar
  4. 4.
    P. Brunovsky, A classification of linear controllable systems, Kybernetica 3 (1970)Google Scholar
  5. 5.
    C. Byrnes, On the control of certain deterministic infinite dimensional systems by algebro-geometric techniques, to appear in Amer.J.Math.Google Scholar
  6. 6.
    C. I. Byrnes and T. E. Duncan, Topological and geometric invariants arising in control theory, to appear.Google Scholar
  7. 7.
    C. Byrnes and N. Hurt, On the moduli of linear dynamical systems, Adv. in Math. Studies in Analysis 4 (1979), 83–122.Google Scholar
  8. 8.
    C. Carathéodory, Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung, Teubner, Leipzig, 1935.Google Scholar
  9. 9.
    A. Cauchy, Calcul des indices des fonctions, J. L'École Polytechnique, 1835, 176–229.Google Scholar
  10. 10.
    M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall, London, 1977.Google Scholar
  11. 11.
    M. H. A. Davis and P. Varaiya, Dynamic programming conditions for partially observable stochastic systems, SIAM J. Control 11 (1973), 226–261.Google Scholar
  12. 12.
    T. E. Duncan, Dynamic programming optimality criteria for stochastic systems in Riemannian manifolds, Appl.Math.Optim. 3 (1977), 191–208.Google Scholar
  13. 13.
    T. E. Duncan, An algebro-geometric approach to estimation and stochastic control for linear pure delay time systems, this volume.Google Scholar
  14. 14.
    A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer.J.Math. 79 (1957), 121–138.Google Scholar
  15. 15.
    W. R. Hamilton, Trans.Roy.Irish Acad. 15 (1828), 69; 16 (1830), 1; 16 (1831), 93; 17 (1837), 1.Google Scholar
  16. 16.
    R. Hermann and C. Martin, Applications of algebraic geometry to systems theory: the McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants, SIAM J. Control Optim. 16 (1978), 743–755.Google Scholar
  17. 17.
    C. Hermite, Sur les nombres des racines d'une équation algébrique comprises entre des limites données, J. Reine Angew. Math. 52 (1856), 39–51.Google Scholar
  18. 18.
    A. Hurwitz, Über die bedingungen unter welchen eine gleichung nur wurzeln mit negativen reelen theilen besitzt, Math.Ann. 46 (1895), 273–284Google Scholar
  19. 19.
    R. E.Kalman, Contributions to the theory of optimal control, Bol.Soc.Mat. Mex. 1960, 102–119.Google Scholar
  20. 20.
    R. E. Kalman, Kronecker invariants and feedback, Ordinary Differential Equations, (L. Weiss, ed.), Academic Press, New York, 1972.Google Scholar
  21. 21.
    E. W. Kamen, On an algebraic theory of systems defined by convolution operators, Math. Systems Theory 9 (1975), 57–74.Google Scholar
  22. 22.
    E. W. Kamen, An operator theory of linear functional differential equations, J. Differential Equations 27 (1978), 274–297.Google Scholar
  23. 23.
    J. L. Lagrange, Mémoire sur la théorie des variations des éléments des planètes, Mém.Cl.Sci.Math.Phys.Inst.France (1808), 1–72.Google Scholar
  24. 24.
    L. H. Loomis and S. Sternberg, Advanced Calculus, Addison-Wesley, Reading, Mass., 1968.Google Scholar
  25. 25.
    V. P. Maslov, Theory of Perturbations and Asymptotic Methods (in Russian) MGU, 1965.Google Scholar
  26. 26.
    J. C. Maxwell, On governors, Proc.Roy.Soc.London 16 (1868), 270–283.Google Scholar
  27. 27.
    D. Quillen, Projective modules over polynomial rings, Invent.Math. 36 (1976), 167–171.Google Scholar
  28. 28.
    R. Rishel, Necessary and sufficient dynamic programming conditions for continuous-time stochastic optimal control, SIAM J. Control 8 (1970), 559–571.Google Scholar
  29. 29.
    H. H. Rosenbrock, State-space and Multivariable Theory, Nelson, London, 1970.Google Scholar
  30. 30.
    E. J. Routh, A treatise on the stability of a given state of motion, Macmillan, London, 1877.Google Scholar
  31. 31.
    J. P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Sém. Dubreil-Pisot, no.23, 1957/58.Google Scholar
  32. 32.
    A. A. Suslin, Projective modules over a polynomial ring are free, Dokl.Akad.Nauk. S.S.S.R 229 (1976) (Soviet Math.Dokl. 17 (1976), 1160–1164).Google Scholar
  33. 33.
    N. S. Williams and V. Zakian, A ring of delay operators with applications to delay-differential systems, SIAM J. Control and Optim. 15 (1977), 247–255.Google Scholar
  34. 34.
    W. M. Wonham, Lecture Notes in Stochastic Control, Center for Dynamical Systems, Brown University 1967.Google Scholar
  35. 35.
    W.M. Wonham, On pole assignment in multi-input controllable linear systems, IEEE Trans Auto. Contr. AC-12 (1967), 660–665.Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • T. E. Duncan
    • 1
    • 2
  1. 1.Institute of Applied MathematicsUniversity of BonnFRG
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA

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