Advertisement

Time-Varying Vector Fields

  • Saber Jafarpour
  • Andrew D. Lewis
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter we consider time-varying vector fields. The ideas in this chapter originate (for us) with the paper of Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978), and are nicely summarised in the more recent book by Agrachev and Sachkov (Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, vol. 87. Springer-Verlag, New York/Heidelberg/Berlin, 2004), at least in the smooth case. A geometric presentation of some of the constructions can be found in the paper of Sussmann (Geometry of Feedback and Optimal Control, pp. 463–557. Dekker Marcel Dekker, New York, 1997), again in the smooth case, and Sussmann also considers regularity less than smooth, e.g., finitely differentiable or Lipschitz. There is some consideration of the real analytic case in Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978), but this consideration is restricted to real analytic vector fields admitting a bounded holomorphic extension to a fixed-width neighbourhood of real Euclidean space in complex Euclidean space. One of our results, the rather nontrivial Theorem 6.25, is that this framework of Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978) is sufficient for the purposes of local analysis. However, our treatment of the real analytic case is global, general, and comprehensive. To provide some context for our novel treatment of the real analytic case, we treat the smooth case in some detail, even though the results are probably mostly known. (However, we should say that, even in the smooth case, we could not find precise statements with proofs of some of the results we give.) We also treat the finitely differentiable and Lipschitz cases, so our theory also covers the “standard” Carathéodory existence and uniqueness theorem for time-varying ordinary differential equations, e.g., Sontag (Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. No. 6 in Texts in Applied Mathematics. Springer-Verlag, New York/Heidelberg/Berlin, 1998), Theorem 54. We also consider holomorphic time-varying vector fields, as these have a relationship to real analytic time-varying vector fields that is sometimes useful to exploit.One of the unique facets of our presentation is that we fully explain the rôle of the topologies developed in Chaps.  3 4, and  5. Indeed, one way to understand the principal results of this chapter is that they show that the usual pointwise—in state and time—conditions placed on vector fields to regulate the character of their flows can be profitably phrased in terms of topologies for spaces of vector fields. While this idea is not entirely new—it is implicit in the approach of Agrachev and Gamkrelidze (Mathematics of the USSR-Sbornik 107(4):467–532, 1978)—we do develop it comprehensively and in new directions.

References

  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T.S.: Manifolds, Tensor Analysis, and Applications, 2 edn. No. 75 in Applied Mathematical Sciences. Springer-Verlag (1988)Google Scholar
  2. 2.
    Agrachev, A.A., Gamkrelidze, R.V.: The exponential representation of flows and the chronological calculus. Mathematics of the USSR-Sbornik 107(4), 467–532 (1978)MathSciNetGoogle Scholar
  3. 3.
    Agrachev, A.A., Sachkov, Y.: Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, vol. 87. Springer-Verlag, New York/Heidelberg/Berlin (2004)Google Scholar
  4. 4.
    Beckmann, R., Deitmar, A.: Strong vector valued integrals (2011). URL http://arxiv.org/abs/1102.1246v1. ArXiv:1102.1246v1 [math.FA]
  5. 5.
    Bogachev, V.I.: Measure Theory, vol. 2. Springer-Verlag, New York/Heidelberg/Berlin (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fernández, C.: Regularity conditions on (LF)-spaces. Archiv der Mathematik. Archives of Mathematics. Archives Mathématiques 54, 380–383 (1990)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fritzsche, K., Grauert, H.: From Holomorphic Functions to Complex Manifolds. No. 213 in Graduate Texts in Mathematics. Springer-Verlag, New York/Heidelberg/Berlin (2002)Google Scholar
  8. 8.
    Gunning, R.C.: Introduction to Holomorphic Functions of Several Variables. Volume I: Function Theory. Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole, Belmont, CA (1990)Google Scholar
  9. 9.
    Hewitt, E., Stromberg, K.: Real and Abstract Analysis. No. 25 in Graduate Texts in Mathematics. Springer-Verlag, New York/Heidelberg/Berlin (1975)Google Scholar
  10. 10.
    Jost, J.: Postmodern Analysis, 3 edn. Universitext. Springer-Verlag, New York/Heidelberg/Berlin (2005)zbMATHGoogle Scholar
  11. 11.
    Mangino, E.M.: (LF)-spaces and tensor products. Mathematische Nachrichten 185, 149–162 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Retakh, V.S.: The subspaces of a countable inductive limit. Soviet Mathematics. Doklady. A translation of the mathematics section of Doklady Akademii Nauk SSSR 11, 1384–1386 (1970)zbMATHGoogle Scholar
  13. 13.
    Saunders, D.J.: The Geometry of Jet Bundles. No. 142 in London Mathematical Society Lecture Note Series. Cambridge University Press, New York/Port Chester/Melbourne/Sydney (1989)Google Scholar
  14. 14.
    Schaefer, H.H., Wolff, M.P.: Topological Vector Spaces, 2 edn. No. 3 in Graduate Texts in Mathematics. Springer-Verlag, New York/Heidelberg/Berlin (1999)Google Scholar
  15. 15.
    Schuricht, F., von der Mosel, H.: Ordinary differential equations with measurable right-hand side and parameter dependence. Tech. Rep. Preprint 676, Universität Bonn, SFB 256 (2000)Google Scholar
  16. 16.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2 edn. No. 6 in Texts in Applied Mathematics. Springer-Verlag, New York/Heidelberg/Berlin (1998)Google Scholar
  17. 17.
    Sussmann, H.J.: An introduction to the coordinate-free maximum principle. In: B. Jakubczyk, W. Respondek (eds.) Geometry of Feedback and Optimal Control, pp. 463–557. Dekker Marcel Dekker, New York (1997)Google Scholar
  18. 18.
    Thomas, G.E.F.: Integration of functions with values in locally convex Suslin spaces. Transactions of the American Mathematical Society 212, 61–81 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wengenroth, J.: Retractive (LF)-spaces. Ph.D. thesis, Universität Trier, Trier, Germany (1995)Google Scholar
  20. 20.
    Whitney, H.: Differentiable manifolds. Annals of Mathematics. Second Series 37(3), 645–680 (1936)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Authors 2014

Authors and Affiliations

  • Saber Jafarpour
    • 1
  • Andrew D. Lewis
    • 1
  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

Personalised recommendations