Sur la topologie de l'espace des systèmes linéaires hamiltoniens anti symétriques accessibles

Phan Nguyen Huynh

Annales de l'Institut Fourier, Tome 44 (1994) no. 3, pp. 967-985.

Dans cet article nous donnons les formes normales des sytèmes linéaires hamiltoniens antisymétriques accessibles $H{A}_{n,m,p}$. Nous construisons une stratification et une décomposition cellulaire analytique de $H{A}_{n,m,p}$, puis nous prouvons que son groupe d’homotopie est isomorphe à celui d’une grassmanienne. Ensuite, nous démontrons que $H{A}_{n,m,p}$ est homotopiquement équivalent à l’espace des systèmes linéaires accessibles. En appliquant ces résultats topologiques, on peut prouver qu’il n’existe pas de paramétrisation continue de tous les systèmes hamiltoniens antisymétriques accessibles si la dimension de l’espace d’entrée est plus grande que 1. En utilisant des travaux de M. Guest et U. Helmke, on peut ainsi donner une démonstration du théorème de périodicité de Bott.

In this paper we construct canonical forms with continue on the strata of the stratification on the space of reachable antisymmetric hamiltonian linear systems $H{A}_{n,m,p}$. We prove that the homology group of $H{A}_{n,m,p}$ is isomorphic to those of the Grassmann manifold. Then we prove that $H{A}_{n,m,p}$ is homotopically equivalent to the space of reachable linear systems.

[1] M. Aigner, Combinatorial theory, Grundlehren der Math. Wissenschaften, 234, Springer, 1979. | MR | Zbl

[2] G. Birkhoff and S. Maclane, A survey of modern algebra, Macmillan, New-York, 1977. | Zbl

[3] A. Borel et A. Heafliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France, 89 (1966), 461-513. | Numdam | Zbl

[4] R. Bott, Lecture on Morse Theory, Mathematische Institut der Universität Bonn, 1960.

[5] R. Bott, The stable homotopy of the classical groups, Annals of Math., 70 (1959), 313-337. | MR | Zbl

[6] R.W. Brockett, Some geometric questions in the theory of linear systems, IEEE Trans. Autom. Control AC., 21 (1976), 449-455. | MR | Zbl

[7] P. Brunovsky, A classification of linear controllable systemes, Kybernetica, 3 (1970), 173-187. | MR | Zbl

[8] C.I. Byrnes, Algebraic and Geometric aspects of the analysis of feedback systems, NATO Advanced Study Institutes Series, Series C-Mathematical and Physical Sciences, vol. 62 : Geometrical Methods for the Theory of linear systems, eds C.I. Byrnes and C. Martin (Pro. of NATO Adv. Stu. In. and AMS summer in App. Math., Harvard University, Cambridge, Mass., June 18-29, 1979), D. Reidel Publishing Company, 1980, p. 83-122. | MR | Zbl

[9] C.I. Byrnes and T.C. Duncan, A note on the topology of space of Hamiltonian transfer functions, Lectures in Appl. Math., vol. 18, p. 7-26, 1980, AMS-NASA-NATO Summer Sem., Havard Univ. Cambridge, Mass.

[10] C.I. Byrnes and T.C. Duncan, On certain topological invariants arising in system theory, from New Directions in Applied Mathematics 13, P. Hilton, G. Young eds, New-York, Springer, Verlag, 29-71, 1981. | Zbl

[11] J. Dieudonné, Foundations of modern analysis, vol. 3, Academic Press, 1972.

[12] M. Guest, Some relationships between homotopy theory and differential geometry, Ph. D. Thesis, Wolfson college, University of Oxford, 1981.

[13] V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Englewood Cliffs., New Jersey, 1974. | MR | Zbl

[14] M. Hazewinkel, Moduli and canonical forms for linear dynamical systems II, The topological case, Math. Systems Theory, 10 (1976/1977), 363-385. | MR | Zbl

[15] M. Hazewinkel, Moduli and canonical forms for linear dynamical systems. III: The algebraic geometry case, Lie Groups: History frontiers and Applications, vol. 7, The 1976 AMES Research Center (NASA) Conference on Geometric Control Theory, C. Martin and R. Hermann eds, p. 291-336, Mat. Sci. Press. | Zbl

[16] M. Hazewinkel, (Fine) Moduli (Space) for linear systems: What are they and What are they good for? NATO Advanced Study Institutes Series, Series C, Mathematical and Physical Sciences, vol. 62: Geometrical Methods for the Theory of linear systems, C.I. Byrnes and C. Martin eds (Proc. of NATO Adv. Stu. In. and AMS summer in App. Math., Harvard University, Cambridge, Mass., June 18-29, 1979), D. Reidel Publishing Company, 1980, p. 125-193. | MR | Zbl

[17] U. Helmke, Zur topologie des Raumes linearer Kontrollsysteme, Ph. D. Thesis, University Bremen, 1982.

[18] D. Hinrichsen, Metrical and topological aspects of linear control theory., Syst. Anal. Model. Simul., 4 (1987), 1, 3-36. | MR | Zbl

[19] D. Hinrichen, D. Salomon, A.J. Pritchard, E.P. Crouch and al., Introduction to Mathematical system theory, Lecture notes for a joint course at the Universities of Warwick and Bremen, 1983. | Zbl

[20] M.W. Hirsch, Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. | MR | Zbl

[21] R.E. Kalman, P.L. Arbib and P.L. Falb, Topics in Mathematical System Theory, McGraw-Hill, 1969. | Zbl

[22] Lê Dung Trang, Sur un critère d'équisingularité, fonctions de plusieurs variables complexes, Lecture Notes in Math., Springer-Verlag, Berlin-Heidelberg, New York, vol. 409 (1974), 124-160. | MR | Zbl

[23] Lê Dung Trang et B. Tessier, Cycles évanescents, sections planes et conditions de Whitney II, Proc. Symposia Pure Math., vol. 40, part 2 (1983), 65-103. | MR | Zbl

[24] B.M. Mann and R.J. Milgram, Some space of holomorphic maps to complex Grassmann manifolds, J. Differential Geometry, 33 (1991), 301-324. | MR | Zbl

[25] W.S. Massey, Homology and homotopy theory, Marcel Dekker, New York, 1978.

[26] J.W. Milnor, Morse theory, Princeton University Press, 1963. | Zbl

[27] J.W. Milnor and J.D. Stasheff, Characteristic classes, Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1974. | MR | Zbl

[28] R. Mneimné et F. Testard, Introduction à la théorie des groupes de Lie classiques, Méthodes, Hermann, Paris, 1986. | MR | Zbl

[29] R. Narasimhan, Analysis on real and complex manifold, Masson CIE, Paris, North-Holland Pub. Comp., Amsterdam, 1968. | MR | Zbl

[30] Nguyen Huynh Phan, On the topology of the space of reachable symmetric linear systems, Math. J. (TAP CHI TOAN HOC), Ha Noi, XV, I (1987), 16-26. | MR | Zbl

[31] Nguyen Huynh Phan, On the topology of the space of reachable symmetric linear systems, Lecture Notes in Mathamatics, vol. 1474, 1991, 235-253, Springer-Verlag, Berlin-Heidelberg, New York (Proccedings of the International Conference on Algebraic Topology, Poznan, Poland, June 22-27, S. Jackowski, B. OLiver, K. Pawalowski eds). | MR | Zbl

[32] Nguyen Huynh Phan, On the topology of the space of reachable skew-symmetric Hamiltonian linear systems, Rendiconti di Matematica, Serie VII, vol. 11, Roma 541-558, 1991. | MR | Zbl

[33] V.M. Popov, Invariant description of linear time-invariant controllable systems, SIAM J. Control, 10 (1972), 252-264. | MR | Zbl

[34] E.H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. | MR | Zbl

[35] G. Segal, The topology of space of rational funtions, Acta Math., 143 (1979), 39-72. | MR | Zbl

[36] H. Whitney, Tangents to an analytic variety, Ann. of Math., (2), 81 (1965), 496-549. | MR | Zbl

[37] J.C. Wonham, Linear multivariable control : A geometric approach, Lecture Notes, Economical and Math. Systems, 101, Springer, 1974. | Zbl