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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.

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 |

### Keywords:

chronological calculus; chronological connection; Lie bracket polynomials; high-order variations; nilpotent polynomial approximation; local controllability
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\textit{A. A. Agrachev} et al., Acta Appl. Math. 14, No. 3, 191--237 (1989; Zbl 0681.49018)

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### References:

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