Local invariants of smooth control systems. (English) Zbl 0681.49018

Summary: Methods are presented for locally studying smooth nonlinear control systems on the manifold \({\mathbb{M}}\). The technique of chronological calculus [see the first two authors, Math. USSR, Sb. 35, 727-785 (1979); translation from Mat. Sb., n. Ser. 107(149), 467-532 (1978; Zbl 0408.34044); J. Sov. Math. 17, 1650-1675 (1981); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 11, 135-176 (1980; Zbl 0473.58021)] is intensively exploited. The concept of chronological connection is introduced and is used when obtaining the invariant expressions in the form of Lie bracket polynomials for high-order variations of a nonlinear control system.
The theorem on adduction of a family of smooth vector fields to the canonical form proved in Section 4 is then applied to the construction of a nilpotent polynomial approximation for a control system. Finally, the relation between the attainable sets of an original system and an approximating one is established; it implies some conclusions on the local controllability of these systems.


49K15 Optimality conditions for problems involving ordinary differential equations
93B03 Attainable sets, reachability
93C10 Nonlinear systems in control theory
Full Text: DOI


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