Introduction to Controllability of Nonlinear Systems

  • Ugo BoscainEmail author
  • Mario SigalottiEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 33)


We present some basic facts about the controllability of nonlinear finite dimensional systems. We introduce the concepts of Lie bracket and of Lie algebra generated by a family of vector fields. We then prove the Krener theorem on local accessibility and the Chow-Rashevskii theorem on controllability of symmetric systems. We then introduce the theory of compatible vector fields and we apply it to study control-affine systems with a recurrent drift or satisfying the strong Lie bracket generating assumption. We conclude with a general discussion about the orbit theorem by Sussmann and Nagano.


Controllability Chow theorem Compatible vector fields Orbit theorem 



This work was supported by the ANR project “SRGI” ANR-15- CE40-0018, by the ANR project “Quaco” ANR-17-CE40-0007-01 and by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH (in a joint call with Programme Gaspard Monge en Optimisation et Recherche Opérationnelle).


  1. 1.
    A.A. Agrachev, Y.L. Sachkov, Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences, vol. 87 (Springer, Berlin, 2004). Control Theory and Optimization, IICrossRefGoogle Scholar
  2. 2.
    F. Jean, Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning. Springer Briefs in Mathematics (Springer, Cham, 2014)Google Scholar
  3. 3.
    V. Jurdjevic, Geometric Control Theory. Cambridge Studies in Advanced Mathematics, vol. 52 (Cambridge University Press, Cambridge, 1997)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CNRS, Team Inria CAGE, Laboratoire Jacques-Louis LionsSorbonne UniversitéParisFrance
  2. 2.Team Inria CAGE, Laboratoire Jacques-Louis LionsSorbonne UniversitéParisFrance

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