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2000 | 74 | 1 | 117-132
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
74
Numer
1
Strony
117-132
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-07-30
poprawiono
2000-02-05
Twórcy
autor
  • Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
  • [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.
  • [C] W. L. Chow, Ueber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98-105.
  • [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.
  • [G] A. M. Gabrielov, Formal relations between analytic functions, Math. USSR-Izv. 37 (1973), 1056-1088.
  • [J1] B. Jakubczyk, Existence and uniqueness of realizations of nonlinear systems, SIAM J. Control Optim. 18 (1980), 455-471.
  • [J2] B. Jakubczyk, Réalisations locales des opérateurs causals non linéaires, C. R. Acad. Sci. Paris 299 (1984), 787-789.
  • [J3] B. Jakubczyk, Local realizations of nonlinear causal operators, SIAM J. Control Optim. 24 (1986), 230-242.
  • [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.
  • [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.
  • [Ma] B. Malgrange, Equation de Lie. II, J. Differential Geometry 7 (1972), 117-141.
  • [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.
  • [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.
  • [N] T. Nagano, Linear differential systems with singularities and applications to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404.
  • [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).
  • [St1] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geometry 24 (1986), 221-263.
  • [St2] R. S. Strichartz, Corrections to 'Sub-Riemannian geometry', ibid. 30 (1989), 595-596.
  • [S1] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188.
  • [S2] H. J. Sussmann, unpublished manuscript.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv74z1p117bwm
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