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has nonempty interior in the orbit topology, as it follows from Krener's theorem applied to the control system restricted to the orbit through
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36:
system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding
704:
204:
1011:
Krener, Arthur J. (1974). "A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems".
818:
As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from
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455:{\displaystyle \ \mathrm {Lie} _{q}\,{\mathcal {F}}=\{g(q)\mid g\in \mathrm {Lie} \,{\mathcal {F}}\}}
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of finite-dimensional control systems. It states that any attainable set of a
40:. Heuristically, Krener's theorem prohibits attainable sets from being
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belongs to the closure of the interior of the attainable set from
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belongs to the closure of the interior of the attainable set from
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291:
210:
756:{\displaystyle \ \mathrm {Lie} _{q}\,{\mathcal {F}}=T_{q}M}
257:{\displaystyle {\mathcal {F}}=\{f(\cdot ,u)\mid u\in U\}}
811:. This is a consequence of Krener's theorem and of the
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585:{\displaystyle \mathrm {Lie} _{q}\,{\mathcal {F}}}
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976:Sussmann, Héctor J.; Jurdjevic, Velimir (1972).
299:{\displaystyle \ \mathrm {Lie} \,{\mathcal {F}}}
926:Agrachev, Andrei A.; Sachkov, Yuri L. (2004).
8:
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929:Control theory from the geometric viewpoint
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201:. Consider the family of vector fields
978:"Controllability of nonlinear systems"
100:{\displaystyle {\ }{\dot {q}}=f(q,u)}
7:
28:about the topological properties of
107:be a smooth control system, where
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694:{\displaystyle \ {\mathcal {F}}}
870:, then the attainable set from
674:When all the vector fields in
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327:{\displaystyle {\mathcal {F}}}
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224:
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82:
1:
1047:Theorems in dynamical systems
1003:10.1016/0022-0396(72)90007-1
336:Lie bracket of vector fields
951:Jurdjevic, Velimir (1997).
1063:
959:Cambridge University Press
625:, the attainable set from
20:is a result attributed to
982:J. Differential Equations
178:belongs to a control set
954:Geometric control theory
840:{\displaystyle \ q\in M}
618:{\displaystyle \ T_{q}M}
545:Remarks and consequences
488:{\displaystyle \ T_{q}M}
360:{\displaystyle \ q\in M}
961:. pp. xviii+492.
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1013:SIAM J. Control Optim
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893:is actually equal to
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125:{\displaystyle {\ q}}
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936:. pp. xiv+412.
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334:with respect to the
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994:1972JDE....12...95S
909:{\displaystyle \ M}
886:{\displaystyle \ q}
863:{\displaystyle \ M}
804:{\displaystyle \ q}
781:{\displaystyle \ q}
664:{\displaystyle \ q}
641:{\displaystyle \ q}
534:{\displaystyle \ q}
511:{\displaystyle \ q}
194:{\displaystyle \ U}
171:{\displaystyle \ u}
148:{\displaystyle \ M}
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34:bracket-generating
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22:Arthur J. Krener
18:Krener's theorem
16:In mathematics,
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1025:10.1137/0312005
1010:
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934:Springer-Verlag
925:
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872:
871:
849:
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701:are analytic,
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30:attainable sets
12:
11:
5:
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1049:
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1042:Control theory
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948:
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764:if and only if
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26:control theory
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988:(1): 95–116.
987:
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968:0-521-49502-4
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943:3-540-21019-9
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813:orbit theorem
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310:generated by
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47:
45:
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39:
35:
31:
27:
24:in geometric
23:
19:
1016:
1012:
985:
981:
953:
928:
847:is dense in
817:
763:
673:
548:
462:is equal to
266:
51:
17:
15:
308:Lie algebra
1036:Categories
920:References
1019:: 43–52.
832:∈
428:∈
422:∣
352:∈
338:. Given
246:∈
240:∣
228:⋅
71:˙
549:Even if
990:Bibcode
495:, then
306:be the
48:Theorem
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61:
52:Let
42:hairy
38:orbit
963:ISBN
938:ISBN
267:Let
155:and
1021:doi
998:doi
1038::
1017:12
1015:.
996:.
986:12
984:.
980:.
957:.
932:.
916:.
815:.
671:.
541:.
264:.
44:.
1027:.
1023::
1006:.
1000::
992::
971:.
946:.
904:M
881:q
858:M
835:M
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799:q
776:q
751:M
746:q
742:T
738:=
733:F
725:q
720:e
717:i
714:L
687:F
659:q
636:q
613:M
608:q
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474:T
450:}
445:F
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410:g
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252:}
249:U
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234:u
231:,
225:(
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189:U
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