Download PDF - Convergence of power series along vector fields and their commutators; a Cartan-Kähler type theoremAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Convergence of power series along vector fields and their commutators; a Cartan-Kähler type theorem

Authors 1

Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

Abstract

We study convergence of formal power series along families of formal or analytic vector fields. One of our results says that if a formal power series converges along a family of vector fields, then it also converges along their commutators. Using this theorem and a result of T. Morimoto, we prove analyticity of formal solutions for a class of nonlinear singular PDEs. In the proofs we use results from control theory.

Keywords

ENG
control systems, Cartan-Kähler theorem, power series, convergence, commutators, Cauchy estimates, vector fields

Bibliography

  1. [CGM] F. Celle, J.-P. Gauthier and E. Milani, Existence of realizations of nonlinear input-output maps, IEEE Trans. Automat. Control. AC-31 (1986), 378-381.
  2. [C] W. L. Chow, Ueber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98-105.
  3. [F] M. Fliess, Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives, Invent. Math. 71 (1983), 521-537.
  4. [G] A. M. Gabrielov, Formal relations between analytic functions, Math. USSR-Izv. 37 (1973), 1056-1088.
  5. [J1] B. Jakubczyk, Existence and uniqueness of realizations of nonlinear systems, SIAM J. Control Optim. 18 (1980), 455-471.
  6. [J2] B. Jakubczyk, Réalisations locales des opérateurs causals non linéaires, C. R. Acad. Sci. Paris 299 (1984), 787-789.
  7. [J3] B. Jakubczyk, Local realizations of nonlinear causal operators, SIAM J. Control Optim. 24 (1986), 230-242.
  8. [J4] B. Jakubczyk, Realizations of nonlinear systems; three approaches, in: Algebraic and Geometric Methods in Nonlinear Control Theory, M. Fliess and M. Hazewinkel (eds.), Reidel, 1986, 3-31.
  9. [K] A. Krener, A generalization of Chow's theorem and the bang-bang theorem to nonlinear control problems, SIAM J. Control 12 (1974), 43-52.
  10. [Ma] B. Malgrange, Equation de Lie. II, J. Differential Geometry 7 (1972), 117-141.
  11. [Mo1] T. Morimoto, Théorème de Cartan-Kähler dans une classe de fonctions formelles Gevrey, C. R. Acad. Sci. Paris Sér. A 311 (1990), 433-436.
  12. [Mo2] T. Morimoto, Théorème d'existence de solutions analytiques pour des systèmes d'équations aux dérivées partielles non-linéaires avec singularités, C. R. Acad. Sci. Paris Sér. I 321 (1995), 1491-1496.
  13. [N] T. Nagano, Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404.
  14. [R] P. K. Rashevskiĭ, On joining two points of a completely nonholonomic space by an admissible curve, Uchen. Zapiski Pedagog. Inst. im. Liebknechta Ser. Fiz.-Mat. 1938, no. 2, 83-94 (in Russian).
  15. [St1] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geometry 24 (1986), 221-263.
  16. [St2] R. S. Strichartz, Corrections to 'Sub-Riemannian geometry', ibid. 30 (1989), 595-596.
  17. [S1] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188.
  18. [S2] H. J. Sussmann, unpublished manuscript.
Annales Polonici Mathematici
2000/74/1
Export to PBN, Opens in new tab
Pages:
117-132
Main language of publication
English
Received
1999-07-30
Accepted
2000-02-05
Published
2000
Exact and natural sciences