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

Phan Nguyen Huynh

Annales de l'Institut Fourier, Tome 44 (1994), p. 967-985 / Harvested from Numdam

Résumé

Dans cet article nous donnons les formes normales des sytèmes linéaires hamiltoniens antisymétriques accessibles $HA_{n,m,p}$. Nous construisons une stratification et une décomposition cellulaire analytique de $HA_{n,m,p}$, puis nous prouvons que son groupe d’homotopie est isomorphe à celui d’une grassmanienne. Ensuite, nous démontrons que $HA_{n,m,p}$ 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.

Abstract

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

Détail

Publié le : 1994-01-01

Zbl 0811.70012

DOI : https://doi.org/10.5802/aif.1422

Bibliographie

[1] M. Aigner, Combinatorial theory, Grundlehren der Math. Wissenschaften, 234, Springer, 1979. | MR 80h:05002 | Zbl 0415.05001

[2] G. Birkhoff and S. Maclane, A survey of modern algebra, Macmillan, New-York, 1977. | Zbl 0365.00006

[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 0102.38502

[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 22 #987 | Zbl 0129.15601

[6] R.W. Brockett, Some geometric questions in the theory of linear systems, IEEE Trans. Autom. Control AC., 21 (1976), 449-455. | MR 57 #9177b | Zbl 0332.93040

[7] P. Brunovsky, A classification of linear controllable systemes, Kybernetica, 3 (1970), 173-187. | MR 44 #1476 | Zbl 0199.48202

[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 82e:93082 | Zbl 0479.93048

[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 0483.93049

[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 50 #1276 | Zbl 0361.57001

[14] M. Hazewinkel, Moduli and canonical forms for linear dynamical systems II, The topological case, Math. Systems Theory, 10 (1976/1977), 363-385. | MR 82h:93021a | Zbl 0396.54037

[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 0368.93007

[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 82g:93024 | Zbl 0481.93023

[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 88k:93003 | Zbl 0676.93038

[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 0565.58027

[20] M.W. Hirsch, Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. | MR 56 #6669 | Zbl 0356.57001

[21] R.E. Kalman, P.L. Arbib and P.L. Falb, Topics in Mathematical System Theory, McGraw-Hill, 1969. | Zbl 0231.49001

[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 50 #13026 | Zbl 0296.14004

[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 86c:32005 | Zbl 0532.32003

[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 93e:55022 | Zbl 0736.54008

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

[26] J.W. Milnor, Morse theory, Princeton University Press, 1963. | Zbl 0108.10401

[27] J.W. Milnor and J.D. Stasheff, Characteristic classes, Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1974. | MR 55 #13428 | Zbl 0298.57008

[28] R. Mneimné et F. Testard, Introduction à la théorie des groupes de Lie classiques, Méthodes, Hermann, Paris, 1986. | MR 89b:22001 | Zbl 0598.22001

[29] R. Narasimhan, Analysis on real and complex manifold, Masson CIE, Paris, North-Holland Pub. Comp., Amsterdam, 1968. | MR 40 #4972 | Zbl 0188.25803

[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 89m:93036 | Zbl 0940.93502

[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 93a:57038 | Zbl 0760.55007

[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 93b:58056 | Zbl 0735.70013

[33] V.M. Popov, Invariant description of linear time-invariant controllable systems, SIAM J. Control, 10 (1972), 252-264. | MR 57 #18925 | Zbl 0251.93013

[34] E.H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. | MR 35 #1007 | Zbl 0145.43303

[35] G. Segal, The topology of space of rational funtions, Acta Math., 143 (1979), 39-72. | MR 81c:55013 | Zbl 0427.55006

[36] H. Whitney, Tangents to an analytic variety, Ann. of Math., (2), 81 (1965), 496-549. | MR 33 #745 | Zbl 0152.27701

[37] J.C. Wonham, Linear multivariable control : A geometric approach, Lecture Notes, Economical and Math. Systems, 101, Springer, 1974. | Zbl 0314.93007