Deux applications de la géométrie locale des diffiétés
Fliess, Michel ; Lévine, Jean ; Martin, Philippe ; Rouchon, Pierre
Annales de l'I.H.P. Physique théorique, Tome 67 (1997), p. 275-292 / Harvested from Numdam
Publié le : 1997-01-01
@article{AIHPA_1997__66_3_275_0,
     author = {Fliess, Michel and L\'evine, Jean and Martin, Philippe A. and Rouchon, Pierre},
     title = {Deux applications de la g\'eom\'etrie locale des diffi\'et\'es},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {67},
     year = {1997},
     pages = {275-292},
     mrnumber = {1456514},
     zbl = {0895.58003},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIHPA_1997__66_3_275_0}
}
Fliess, Michel; Lévine, Jean; Martin, Philippe; Rouchon, Pierre. Deux applications de la géométrie locale des diffiétés. Annales de l'I.H.P. Physique théorique, Tome 67 (1997) pp. 275-292. http://gdmltest.u-ga.fr/item/AIHPA_1997__66_3_275_0/

[1] D.V. Alekseevskij, A.M. Vinogradov et V.V. Lychagin, Geometry I, Basic Ideas and Concepts of Differential Geometry, Encycl. Math. Sci., Vol. 28, Springer-Verlag, Berlin, 1991. | MR 1300019

[2] R.L. Anderson et N.H. Ibragimov, Lie-Bäcklund Transformations in Applications, S.I.A.M., Philadelphie, 1979. | Zbl 0447.58001

[3] P. Appell, Traité de Mécanique Rationnelle, t. 2, 6e éd., Gauthier-Villars, Paris, 1953.

[4] V.I. Arnold, V.V. Kozlov et A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encycl. Math. Sci., Vol. 3, Springer-Verlag, Berlin, 1988. | MR 923953

[5] J.E. Björk, Rings of Differential Operators, North-Holland, Amsterdam, 1979. | MR 549189 | Zbl 0499.13009

[6] K.E. Brenan, S.L. Campbell et L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier, Amsterdam, 1989. | MR 1101809 | Zbl 0699.65057

[7] A. Buium, Differential Algebraic Groups of Finite Dimension, Springer-Verlag, Berlin, 1992. | MR 1176753 | Zbl 0756.14028

[8] A. Buium, Differential Algebra and Diophantine Geometry, Hermann, Paris, 1994. | MR 1487891 | Zbl 0870.12007

[9] P.J. Cassidy, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, J. Algebra, Vol. 121, 1989, p. 169-238. | MR 992323 | Zbl 0678.14011

[10] P.M. Cohn, Free Rings and their Relations, 2nd ed., Academic Press, Londres, 1985.

[11] P. Dazord, Mécanique hamiltonienne en présence de contraintes, Illinois J. Math., Vol. 38, 1994, p. 148-175. | MR 1245839 | Zbl 0790.58018

[12] E. Delaleau et W. Respondek, Lowering the orders of derivatives of controls in generalized state space systems, J. Math. Systems Estimat. Control, Vol. 5, 1995, p. 375-378. | MR 1651823 | Zbl 0852.93016

[13] L.A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapour, 1991. | MR 1147643 | Zbl 0753.35075

[14] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester, 1993. | MR 1237398

[15] M. Fliess, Automatique et corps différentiels, Forum Math., Vol. 1, 1989, p. 227-238. | MR 1005424 | Zbl 0701.93048

[16] M. Fliess, Generalized controller canonical forms for linear and nonlinear dynamics, IEEE Trans. Automat. Control, Vol. 35, 1990, p. 994-1001. | MR 1065035 | Zbl 0724.93010

[17] Fliess and S.T. Glad, An algebraic approach to linear and nonlinear control, in Essays on Control: Perspectives in the Theory and its Applications (H. Trentelman and J. C. Willems Eds), Birkhäuser, Boston, 1993, p. 223-267. | MR 1247273 | Zbl 0838.93021

[18] M. Fliess, J. Lévine, P. Martin, F. Ollivier et P. Rouchon, Flatness and dynamic feedback linearizability: two approaches, Proc. 3rd European Control Conf., Rome, 1995, p. 649-654.

[19] M. Fliess, J. Lévine, P. Martin et P. Rouchon, Linéarisation par bouclage dynamique et transformations de Lie-Bäcklund, C. R. Acad. Sci. Paris, Vol. I-317, 1993, p. 981-986. | MR 1249373 | Zbl 0796.93042

[20] M. Fliess, J. Lévine, P. Martin et P. Rouchon, Nonlinear control and Lie-Bäcklund transformations: Towards a new differential geometric standpoint, Proc. IEEE Control Decision Conf., Lake Buena Vista, FL, 1994, p. 339-344.

[21] M. Fliess, J. Lévine, P. Martin et P. Rouchon, Flatness and defect of non-linear systems: introductory theory and examples, Internat. J. Control, Vol. 61, 1995, p. 1327-1361. | MR 1613557 | Zbl 0838.93022

[22] M. Fliess, J. Lévine, P. Martin et P. Rouchon, Index and decomposition of nonlinear implicit differential equations, Proc. IFAC Conf. System Structure Control, Nantes, 1995, p. 43-48.

[23] M. Fliess, J. Lévine, P. Martin et P. Rouchon, Design of trajectory stabilizing feedback for driftless flat systems, Proc. 3rd European Control Conf., Rome, 1995, p. 1882-1887.

[24] M. Fliess, J. Lévine, P. Martin et P. Rouchon, A Lie-Bäcklund transformation approach to equivalence and flatness of nonlinear systems, à paraître.

[25] M. Fliess, J. Lévine et P. Rouchon, A generalised state variable representation for a simplified crane description, Internat. J. Control, Vol. 58, 1993, p. 277-283. | MR 1229850 | Zbl 0782.93049

[26] G. Giachetta, Jet methods in nonholonomic mechanics, J. Math. Physics, Vol. 33, 1992, p. 1652-1665. | MR 1158984 | Zbl 0758.70010

[27] M. Gromov, Partial Differential Relations, Springer-Verlag, Berlin, 1986. | MR 864505 | Zbl 0651.53001

[28] V.N. Gusyatnikova, A.M. Vinogradov et V.A. Yumaguzhin, Secondary differential operators, J. Geometry Physics, Vol. 2, 1985, p. 23-65. | MR 845467 | Zbl 0601.35007

[29] E. Hairer, C. Lubich et M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Springer-Verlag, Berlin, 1989. | MR 1027594 | Zbl 0683.65050

[30] G. Hamel, Theoretische Mechanik, Springer-Verlag, Berlin, 1949. | MR 516809

[31] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. | MR 463157 | Zbl 0367.14001

[32] M. Henneaux et C. Teitelboim, Quantization of Gauge Systems, Princeton University Press, Princeton, 1992. | MR 1191617 | Zbl 0838.53053

[33] N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. | Zbl 0558.53040

[34] A. Isidori, Nonlinear Control Systems, 3rd ed., Springer-Verlag, New York, 1995.

[35] J. Johnson, Kähler differentials and differential algebra, Ann. of Math., Vol. 89, 1969, p. 92-98. | MR 238823 | Zbl 0179.34302

[36] J. Johnson, Prolongations of integral domains, J. Algebra, Vol. 94, 1985, p. 173-211. | MR 789546

[37] E.R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. | MR 568864 | Zbl 0264.12102

[38] E.R. Kolchin, Differential Algebraic Groups, Academic Press, Orlando, 1985. | MR 776230 | Zbl 0556.12006

[39] I.S. Krasil'Shchik, V.V. Lychagin et A.M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986. | MR 861121 | Zbl 0722.35001

[40] B.A. Kupershmidt, The Variational Principles of Dynamics, World Scientific, Singapour, 1992. | MR 1207138 | Zbl 0917.58001

[41] Yu.I. Manin, Algebraic aspects of nonlinear differential equations, J. Soviet Math., Vol. 11, 1979, p. 1-122. | Zbl 0419.35001

[42] E. Massa et E. Pagani, Classical dynamics of non-holonomic systems, Ann. Inst. H. Poincaré Phys. Théor., Vol. 55, 1991, p. 511-544. | Numdam | MR 1130215 | Zbl 0731.70012

[43] E. Massa et E. Pagani, Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics, Ann. Inst. H. Poincaré Phys. Théor., Vol. 61, 1994, p. 17-62. | Numdam | MR 1303184 | Zbl 0813.70004

[44] Ju.I. Neĭmark et N.A. Fufaev, Dynamics of Nonholonomic Systems, Amer. Math. Soc., Providence, R.I., 1972.

[45] H. Nijmeijer et Van Der Schaft, Nonlinear Control Systems, Springer-Verlag, New York, 1990. | Zbl 0701.93001

[46] P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, New York, 1993. | MR 1240056 | Zbl 0785.58003

[47] J.F. Ritt, Differential Algebra, Amer. Math. Soc., New York, 1950. | Zbl 0037.18402

[48] J.C. Tougeron, Idéaux de Fonctions Différentiables, Springer-Verlag, Berlin, 1972. | MR 440598 | Zbl 0251.58001

[49] T. Tsujishita, Formal geometry of systems of differential equations, Sugaku Expos., Vol. 3, 1990, p. 25-73. | Zbl 0713.58020

[50] T. Tsujishita, Homological method of computing invariants of systems of differential equations, Diff. Geometry Appl., Vol. 1, 1991, p. 3-34. | MR 1109811 | Zbl 0722.58036

[51] A.M. Vinogradov, Local symmetries and conservation laws, Acta Appl. Math., Vol. 2, 1984, p. 21-78. | MR 736872 | Zbl 0534.58005

[52] A.M. Vinogradov, Ed., Symmetries of Partial Differential Equations, Kluwer, Dordrecht, 1989 (reprinted from Acta Appl. Math., Vol. 15, 1989, n° 1-2, and Vol. 16, 1989, n° 1-2). | MR 1007340 | Zbl 0684.35002

[53] A.M. Vinogradov, Scalar differential invariants, diffieties and characteristic classes, in Mechanics, Analysis and Geometry: 200 Years after Lagrange (M. Francaviglia Ed.), North-Holland, Amsterdam, 1991, p. 379-416. | MR 1098525 | Zbl 0735.57012

[54] A.M. Vinogradov, From symmetries of partial differential equations towards secondary ("quantized") calculus, J. Geometry Physics, Vol. 14, 1994, p. 146-194. | MR 1288219 | Zbl 0815.58028

[55] E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., Cambridge University Press, Cambridge, 1947. | MR 992404

[56] V.V. Zharinov, On Bäcklund correspondences, Math. USSR Sb., Vol. 64, 1989, p. 277-293. | MR 954929 | Zbl 0683.35005

[57] V.V. Zharinov, Geometrical Aspects of Partial Differential Equations, World Scientific, Singapour, 1992. | MR 1167448 | Zbl 0763.58002