50.0.193.12: Add some links

2014-08-23T05:50:45Z

Add some links

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en>Wavelength: inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

2014-02-20T20:11:51Z

inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

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en>Ebony Jackson: trivial ring --> zero ring

2013-12-14T07:27:24Z

trivial ring --> zero ring

← Older revision		Revision as of 09:27, 14 December 2013
Line 1:		Line 1:
	~~Hi there~~. Let ~~me start by introducing~~ the ~~writer~~, ~~her name is Sophia Boon but she~~ by ~~no means truly favored that name~~. ~~She~~ is ~~truly fond~~ of ~~caving but she doesn't have~~ the ~~time recently~~. ~~For many years he~~'s ~~been living~~ in ~~Alaska~~ and ~~he doesn~~'~~t plan on altering it~~. ~~I am an invoicing officer~~ and I'~~ll be promoted quickly.~~<br><br>~~My weblog good psychic (~~[http://~~formalarmour~~.com/~~index~~.~~php~~?do=/~~profile~~-~~26947/info/ formalarmour~~.~~com~~])		In mathematics, '''Krener's theorem''' is a result in geometric [[control theory]] about the topological properties of [[attainable set]]s of finite-dimensional control systems. It states that any attainable set of a [[bracket-generating]] system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding [[orbit (control theory)\|orbit]]. Heuristically, Krener's theorem prohibits attainable sets from being [[hairy ball theorem\|hairy]].

			==Theorem==
			Let
			<math>{\ }\dot q=f(q,u)</math>
			be a smooth control system, where
			<math>{\ q}</math>
			belongs to a <!-- <math>\ n</math>- -->finite-dimensional manifold <math>\ M</math> and <math>\ u</math> belongs to a control set <math>\ U</math>. Consider the family of vector fields <math>{\mathcal F}=\{f(\cdot,u)\mid u\in U\}</math>.

			Let <math>\ \mathrm{Lie}\,\mathcal{F}</math> be the [[Lie algebra]] generated by <math>{\mathcal F}</math> with respect to the [[Lie bracket of vector fields]].
			Given <math>\ q\in M</math>, if the vector space <math>\ \mathrm{Lie}_q\,\mathcal{F}=\{g(q)\mid g\in \mathrm{Lie}\,\mathcal{F}\}</math> is equal to <math>\ T_q M</math>,
			then <math>\ q</math> belongs to the closure of the interior of the attainable set from <math>\ q</math>.

			==Remarks and consequences==
			Even if <math>\mathrm{Lie}_q\,\mathcal{F}</math> is different from <math>\ T_q M</math>,
			the attainable set from <math>\ q</math> has nonempty interior in the orbit topology,
			as it follows from Krener's theorem applied to the control system restricted to the orbit through <math>\ q</math>.

			When all the vector fields in <math>\ \mathcal{F}</math> are analytic, <math>\ \mathrm{Lie}_q\,\mathcal{F}=T_q M</math> '''if and only if''' <math>\ q</math> belongs to the closure of the interior of the attainable set from <math>\ q</math>. This is a consequence of Krener's theorem and of the [[orbit (control theory)#Orbit theorem (Nagano-Sussmann)\|orbit theorem]].

			As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from <math>\ q\in M</math> is dense in <math>\ M</math>, then the attainable set from <math>\ q</math>
			is actually equal to <math>\ M</math>.

			==References==

			*{{cite book
			\| last = Agrachev
			\| first = Andrei A.
			\| authorlink =
			\| coauthors = Sachkov, Yuri L.
			\| title = Control theory from the geometric viewpoint
			\| publisher = [[Springer-Verlag]]
			\| date =2004
			\| location =
			\| pages = xiv+412
			\| url =http://www.springer.com/east/home/math/applications?SGWID=5-10051-22-26872681-0
			\| doi =
			\| id =
			\| isbn = 3-540-21019-9}}
			*{{cite book
			\| last =Jurdjevic
			\| first =Velimir
			\| authorlink =
			\| coauthors =
			\| title =Geometric control theory
			\| publisher =[[Cambridge University Press]]
			\| date =1997
			\| location =
			\| pages =xviii+492
			\| url =http://www.cup.cam.ac.uk/us/catalogue/email.asp?isbn=9780521495028
			\| doi =
			\| id =
			\| isbn = 0-521-49502-4}}
			*{{cite journal
			\| last =Sussmann
			\| first =Héctor J.
			\| authorlink =
			\| coauthors =Jurdjevic, Velimir
			\| title =Controllability of nonlinear systems
			\| journal =J. Differential Equations
			\| volume =12
			\| issue =
			1\| pages =95–116
			\| publisher =
			\| location =
			\| date =1972
			\| url =
			\| doi =10.1016/0022-0396(72)90007-1
			\| id =
			\| accessdate = }}
			*{{cite journal
			\| last =Krener
			\| first =Arthur J.
			\| authorlink =
			\| coauthors =
			\| title =A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems
			\| journal =SIAM J. Control Optim.
			\| volume =12
			\| issue =
			\| pages =43–52
			\| publisher =
			\| location =
			\| date =1974
			\| url =
			\| doi =
			\| id =
			\| accessdate = }}

			[[Category:Control theory]]
			[[Category:Theorems in dynamical systems]]

en>PS745 at 11:49, 17 April 2012

2012-04-17T11:49:25Z

New page

Hi there. Let me start by introducing the writer, her name is Sophia Boon but she by no means truly favored that name. She is truly fond of caving but she doesn't have the time recently. For many years he's been living in Alaska and he doesn't plan on altering it. I am an invoicing officer and I'll be promoted quickly.<br><br>My weblog good psychic ([http://formalarmour.com/index.php?do=/profile-26947/info/ formalarmour.com])

Field with one element - Revision history

50.0.193.12: Add some links

en>Wavelength: inserting 1 hyphen: —> "one-dimensional"—wikt:one-dimensional

en>Ebony Jackson: trivial ring --> zero ring

en>PS745 at 11:49, 17 April 2012