|
Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
Retrieve article in:
PDF
Book Information
Author(s):
Robert Hermann
Title:
Cartanian geometry, nonlinear waves, and control theory
Additional book information:
Interdisciplinary Mathematics, volumes 20 and 21, Math. Sci. Press, Brookline, Mass., Part A, 1979, xv + 501 pp., $50.00; Part B, 1980, xii + 585 pp., $60.00
References:
- 1.
- R. Abraham and J. Marsden, Foundations of mechanics, 2nd ed., Benjamin-Cummings, Reading, Mass., 1978. MR 515141
- 2.
- V. I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Math., No. 60, Springer-Verlag, New York, 1978. MR 690288
- 3.
- K. Bleuler and A. Reetz (Editors), Differential geometric methods in mathematical physics. I, Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin-Heidelberg, 1977. MR 434688
- 4.
- K. Bleuler, H. R. Petry and A. Reetz (Editors), Differential geometric methods in mathematical physics. II, Lecture Notes in Math., vol. 676, Springer-Verlag, Berlin-Heidelberg, 1978. MR 519605
- 5.
- F. Bloom, Modern differential geometric techniques in the theory of continuous distributions of dislocations, Lecture Notes in Math., vol. 733, Springer-Verlag, Berlin, 1979. MR 546839
- 6.
- R. W. Brockett, Finite dimensional linear systems, Wiley, New York, 1969.
- 7.
- R. W. Brockett, Systems theory on group manifolds and coset spaces, SIAM J. Control Optim. 10 (1972), 265-284. MR 315559
- 8.
- R. W. Brockett, Lie theory and control systems defined on spheres, SIAM J. Appl. Math. 25 (1973), 213-225. MR 327337
- 9.
- R. W. Brockett, Nonlinear systems and differential geometry, Proc. of the IEEE 64 (1976), 61-72. MR 432255
- 10.
- P. Brunovsky, A classification of linear controllable systems, Kybernetika (Prague) 6 (3), 1970, 173-188. MR 284247
- 11.
- C. I. Byrnes (Editor), Partial differential equations and geometry, Proc. Park City Conf., Lecture Notes in Pure and Appl. Math., Vol. 48, Marcel Dekker, 1979. MR 535586
- 12.
- C. I. Byrnes and C. F. Martin (Editors), Geometrical methods for the theory of linear systems, NATO Advanced Study Institutes Series D, Reidel, Dordrecht, Holland, 1980. MR 608990
- 13.
- P. Chernoff and J. Marsden, Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Math., vol. 425, Springer-Verlag, New York, 1974. MR 650113
- 14.
- P. Chernoff and J. Marsden, Review of "Lie algebras and quantum mechanics" and "Vector bundles in mathematical physics" by Robert Hermann, Bull. Amer. Math. Soc. 79(1973), 1150-1162.
- 15.
- P. E. Crouch, Dynamical realizations of finite Volterra series, SIAM J. Control Optim. 19 (1981), 177-202. MR 605615
- 16.
- M. Daniel and C. M. Viallet, The geometrical setting of gauge theories of the Yang-Mills type, Rev. Modern Phys. 52 (1980), 175-197. MR 553409
- 17.
- E. Dierker, Topological methods in Walrasian economics, Lecture Notes in Economics and Systems, No. 92, Springer-Verlag, Berlin, 1970. MR 378736
- 18.
- J. Dieudonné, Treatise on analysis, vol. 10-IV, Academic Press, New York, 1974. MR 362066
- 19.
- W. Drechsler and M. E. Mayer, Fiber bundle techniques in gauge theories, Lecture Notes in Physics, No. 67, Springer-Verlag, Berlin-Heidelberg, 1977. MR 443737
- 20.
- E. J. Flaherty, Hermitian and Kählerian geometry in relativity, Lecture Notes in Phys., No. 46, Springer-Verlag, Berlin-Heidelberg, 1976. MR 411532
- 21.
- E. J. Flaherty, Complex variables in relativity, General Relativity and Gravitation, Vol. 2 (A. Held, ed.), Plenum Press, New York, 1980, pp. 207-239. MR 617921
- 22.
- T. Frankel, Gravitational curvature, an introduction to Einstein's theory, Freeman, San Francisco, 1979. MR 518868
- 23.
- F. R. Gantmacher, The theory of matrices, Vol. 2, Chelsea, New York, 1959.
- 24.
- R. Gunning, Lectures on vector bundles over Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1967. MR 230326
- 25.
- V. Guillemin and S. Sternberg, Geometric asymptotics, Math. Surveys, no. 14, Amer. Math. Soc. Providence, R. I., 1977. MR 516965
- 26.
- S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Univ. Press, London and New York, 1973. MR 424186
- 27.
- G. W. Haynes and H. Hermes, Nonlinear controllability via Lie theory, SIAM J. Control Optim. 8 (1970), 450-460. MR 277859
- 28.
- R. Hermann, Algebraic topics in systems theory, Interdisciplinary Mathematics, Vol. 3, Math-Sci Press, Brookline, Mass., 1973. MR 504083
- 29.
- R. Hermann, On the accessibility problem in control theory, International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 325-332. MR 149402
- 30.
- R. Hermann, Vector bundles in mathematical physics, vol. 1, Benjamin, New York, 1970. MR 266551
- 31.
- R. Hermann and A. J. Krener, Nonlinear controllability and observability, IEEE Trans. Autom. Control. AC-22 (1977), 728-740. MR 476017
- 32.
- R. Hermann and C. F. Martin, Applications of algebraic geometry to systems theory--Part I, IEEE Trans. Automat. Control AC-22 (1977), 19-25. MR 444172
- 33.
- R. Hermann, Review of "Elements of soliton theory" by G. L. Lamb, Jr., Bull. Amer. Math. Soc. (N. S.) 5 (1981), 203-209.
- 34.
- H. Hermes, On local and global controllability, SIAM J. Control Optim. 12 (1974), 252-261. MR 451293
- 35.
- R. M. Hirschorn, Controllability in nonlinear systems, J. Differential Equations 19 (1975), 46-61. MR 431271
- 36.
- R. M. Hirschorn, Invertibility of control systems on Lie groups, SIAM J. Control Optim. 15 (1977), 1034-1049. MR 527880
- 37.
- R. M. Hirschorn, (A, B)-invariant distributions and disturbance decoupling of nonlinear systems, SIAM J. Control Optim. 19 (1981), 1-19. MR 603076
- 38.
- V. Jurdjevic and H. J. Sussmann, Control systems on Lie groups, J. Differential Equations 12 (1972), 313-329. MR 331185
- 39.
- V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: Accessibility, J. Differential Equations 39 (1981), 186-211. MR 607781
- 40.
- R. E. Kalman, Kronecker invariants and feedback, Conf. Ordinary Differential Equations (NRL Mathematics Research Center, June 14-23, 1971), Ordinary Differential Equations (C. Weiss, ed.), Academic Press, New York, 1972, pp. 459-471. MR 421751
- 41.
- B. Kostant, Quantization and unitary representations, Lecture Notes in Math., vol. 170, Springer-Verlag, New York, 1970. MR 294568
- 42.
- A. J. Krener, A decomposition theory for differentiable systems, SIAM J. Control Optim. 15 (1977), 813-829. MR 479481
- 43.
- A. J. Krener, A. Isidori, C. Gori-Giorgi and S. Monaco, Nonlinear decoupling via feedback: A differential geometric approach, IEEE Trans. Automat. Control, AC-26 (1981), 331-345. MR 613540
- 44.
- D. E. Lerner and P. D. Sommers (Editors), Complex manifold techniques in theoretical physics, Pitman Advanced Publishing Program, London, 1979; Research Notes in Math. 32. MR 564439
- 45.
- C. Lobry, Controlabilite des systemes non lineaires, SIAM J. Control Optim. 8 (1970), 573-605. MR 271979
- 46.
- S. Mac Lane, Geometrical mechanics (2 parts), Dept. of Math., Univ. of Chicago, Illinois, 1968.
- 47.
- J. Marsden, Applications of global analysis in mathematical physics, Publish or Perish, Boston, 1974. MR 646816
- 48.
- C. F. Martin and R. Hermann, Applications of algebraic geometry to systems theory. Part II: Feedback and pole placement for linear Hamiltonian systems, Proc. of the IEEE 65 (1977), 841-848. MR 527293
- 49.
- C. F. Martin and R. Hermann, 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. MR 527294
- 50.
- D. Q. Mayne and R. W. Brockett (Editors), Geometric methods in systems theory, Reidel, Dordrecht, Holland, 1973.
- 51.
- N. D. Mermin, The topological theory of defects in ordered media, Rev. Modern Phys. 51 (1979), 591-648. MR 541885
- 52.
- C. Misner, K. Thorne and J. A. Wheeler, Gravitation, Freeman, San Francisco, 1973. MR 418833
- 53.
- A. I. Murdoch and H. Cohen, Symmetry considerations for material surfaces, Arch. Rational Mech. Anal. 72 (1979), 61-98. MR 540222
- 54.
- W. Noll, Materially uniform simple bodies with inhomogeneitics, Arch. Rational Mech. Anal. 27(1967), 1-32. MR 225530
- 55.
- B. N. Parlett and W. G. Poole, Jr., A geometric theory for the QR, LU and power iterations, SIAM J. Numer. Anal. 10 (1973), 389-412. MR 336979
- 56.
- R. Penrose and R. S. Ward, Twisters for flat and curved space-time, General Relativity and Gravitation, Vol. 2 (A. Held, ed.), Plenum Press, New York, 1980, pp. 283-328. MR 617923
- 57.
- V. Poénaru, Some aspects of the theory of defects of ordered media and gauge fields related to foliations, Comm. Math. Phys. 80(1981), 127-136. MR 623154
- 58.
- R. Sachs and H. Wu, General relativity for mathematicians, Graduate Texts in Math., No. 48, Springer-Verlag, Berlin-Heidelberg and New York, 1977. MR 503498
- 59.
- C. R. Schneider, Global aspects of the matrix Riccati equation, Math. Systems Theory 7 (1973), 281-286. MR 391175
- 60.
- D. J. Simms and N. M. J. Woodhouse, Lectures on geometric quantization, Lecture Notes in Phys., No. 53, Springer-Verlag, Berlin-Heidelberg, 1976. MR 672639
- 61.
- J. Sniatycki, Geometric quantization and quantum mechanics, Applied Mathematical Sciences Series, No. 30, Springer-Verlag, New York, 1980. MR 554085
- 62.
- C. P. Snow, The search, 2nd ed., Schribner, New York, 1958, pp. 203-209.
- 63.
- J. M. Souriau, Structure des systemes dynamiques, Dunod, Paris, 1970. MR 260238
- 64.
- S. Sternberg, Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J., 1963. MR 193578
- 65.
- S. Sternberg, Review of "Foundations of mechanics" by Ralph Abraham and Jerrold E. Marsden, Bull. Amer. Math. Soc. (N. S.) 2 (1980), 378-387.
- 66.
- H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95-116. MR 338882
- 67.
- H. J. Sussmann, Existence and uniqueness of minimal realizations of nonlinear systems, Math. Systems Theory 10 (1977), 263-284. MR 437158
- 68.
- H. J. Sussmann, Subanalytic sets and feedback control, J. Differential Equations 31 (1979), 31-52. MR 524816
- 69.
- W. Thirring, A course in mathematical physics. I, Classical dynamical systems, Springer-Verlag, New York, 1978. MR 507189
- 70.
- W. Thirring, A course in mathematical physics. II, Classical field theory, Springer-Verlag, New York, 1979. MR 553112
- 71.
- A. Trautman, Fiber bundles, gauge fields and gravitation, General Relativity and Gravitation, Vol. 1 (A. Held, ed.), Plenum Press, New York, 1980, pp. 287-308. MR 583721
- 72.
- N. R. Wallach, Symplectic geometry and Fourier analysis, Lie Groups: History, Frontiers and Applications, Vol. V, Math-Sci Press, Brookline, Mass., 1977. MR 488148
- 73.
- C. -C. Wang, On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal. 27 (1967), 33-94. MR 229420
- 74.
- R. S. Ward and L. P. Hughston (Editors), Advances in twistor theory, Pitman Advanced Publishing Program, London, 1979; Research Notes in Math. 37. MR 578487
- 75.
- R. O. Wells, Complex manifolds and mathematical physics, Bull. Amer. Math. Soc. (N. S.) 1 (1979), 296-336. MR 520077
- 76.
- A. Weinstein, Lectures on symplectic manifolds, CBMS Conf. Ser. in Math., No. 29, Amer. Math. Soc., Providence, R. I., 1977. MR 464312
- 77.
- W. M. Wonham, Linear multivariable control: A geometric approach, 2nd ed., Springer-Verlag, New York, 1979. MR 569358
Additional Information:
Reviewer(s):
Clark R.
Givens
Reviewer(s):
Richard S.
Millman
Review Information:
Journal:
Bull. Amer. Math. Soc.
6
(1982),
467-478.
DOI:
10.1090/S0273-0979-1982-15019-4
PII:
S 0273-0979(1982)15019-4
|