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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.

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