Introduction to Controllability of Nonlinear Systems
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.
KeywordsControllability 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).
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