Vector notation - Revision history

en>Fgnievinski: /* See also */

2014-10-05T15:36:18Z

See also

← Older revision		Revision as of 17:36, 5 October 2014
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	~~In [[optimal control]], problems of '''singular control''' are problems~~ that ~~are difficult to solve because~~ a ~~straightforward application of [[Pontryagin~~'~~s minimum principle]] fails to yield a complete solution. Only a few such problems have~~ been ~~solved, such as [[Merton's portfolio problem]] in [[financial economics]] or [[trajectory optimization]] in aeronautics. A more technical explanation follows.~~		Andera is what you can call her but she by no means really favored that title. Doing ballet is something she would never give up. For a whilst I've been in Mississippi but now I'm contemplating other choices. My working day job is an invoicing officer but I've already applied for an additional clairvoyants - [http://www.article-galaxy.com/profile.php?a=143251 www.article-galaxy.com] - 1.<br><br>Also visit my web-site :: accurate [http://si.dgmensa.org/xe/index.php?document_srl=48014&mid=c0102 love psychic] predictions [[http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26]]

	~~The most common difficulty~~ in ~~applying Pontryagin~~'~~s principle arises when the Hamiltonian depends linearly on the control <math>u</math>, i.e~~., is ~~of the form~~: ~~<math>H(u)=\phi(x,\lambda,t)u+\cdots<~~/~~math> and the control is restricted to being between an upper and a lower bound: <math>a\le u(t)\le b<~~/~~math>~~. ~~To minimize <math>H(u)<~~/~~math>, we need to make <math>u</math> as big or as small as possible, depending on the sign of <math>\phi(x,\lambda,t)</math>, specifically:~~

	~~: <math>u(t) = \begin{cases} b, & \phi(x,\lambda,t)<0 \\~~ ?~~, & \phi(x,\lambda,t)~~=~~0 \\ a, & \phi(x,\lambda,t)>0~~.~~\end{cases}</math>~~

	~~If <math>\phi</math> is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a [[bang~~-~~bang control~~]~~] that switches from~~ <~~math~~>b<~~/math~~> ~~to <math>a<~~/~~math> at times when <math>\phi<~~/~~math> switches from negative to positive~~.

	~~The case when <math>\phi</math> remains at zero for a finite length of time <math>t_1\le t\le t_2</math> is called the '''singular control''' case~~. ~~Between <math>t_1<~~/~~math> and <math>t_2<~~/~~math> the maximization of the Hamiltonian with respect to u gives us no useful information and the solution in that time interval is going to have to be found from other considerations~~. ~~(One approach would be to repeatedly differentiate <math>\partial H~~/~~\partial u<~~/~~math> with respect to time until the control u again explicitly appears, which is guaranteed to happen eventually~~. ~~One can then set that expression to zero and solve for u. This amounts to saying that between <math>t_1</math> and <math>t_2</math> the control <math>u<~~/~~math> is determined by the requirement that the singularity condition continues to hold~~. ~~The resulting so-called singular arc will be optimal if it satisfies the '''Kelley condition'''~~:

	~~:<math>(-1)^k \frac{\partial}{\partial u} \left[ {\left( \frac{d}{dt} \right)}^{2k} H_u \right] \ge 0 ,\, k=0,1,\cdots<~~/~~math>~~

	.~~<ref>Bryson, Ho: Applied Optimal Control, Page 246<~~/~~ref> This condition is also called the generalized [[Legendre-Clebsch condition]])~~.

	~~The term '''bang-singular control''' refers to a control that has a bang-bang portion as well as a singular portion.~~

	~~==References~~==

	~~{{reflist}}~~

	~~{{DEFAULTSORT:Singular Control}}~~
	~~[[Category:Control theory~~]]

en>Mvakkur: tidied up the language a bit, removed some wordy phrases, deleted some unnecessary commas

2013-11-13T20:05:24Z

tidied up the language a bit, removed some wordy phrases, deleted some unnecessary commas

← Older revision		Revision as of 22:05, 13 November 2013
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	~~Hi there~~. ~~My title~~ is ~~Sophia Meagher even though it~~ is ~~not~~ the ~~name on my beginning certification~~. ~~To climb~~ is some ~~thing I truly appreciate performing~~. ~~Alaska~~ is ~~exactly where I~~'~~ve usually been living. My working day occupation is an information officer but I~~'~~ve currently applied for another one~~.<br><br>~~My web page; telephone psychic~~ (~~[http:~~//~~appin~~.co.kr/~~board_Zqtv22~~/~~688025 appin~~.co.kr])		In [[optimal control]], problems of '''singular control''' are problems that are difficult to solve because a straightforward application of [[Pontryagin's minimum principle]] fails to yield a complete solution. Only a few such problems have been solved, such as [[Merton's portfolio problem]] in [[financial economics]] or [[trajectory optimization]] in aeronautics. A more technical explanation follows.

			The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control <math>u</math>, i.e., is of the form: <math>H(u)=\phi(x,\lambda,t)u+\cdots</math> and the control is restricted to being between an upper and a lower bound: <math>a\le u(t)\le b</math>. To minimize <math>H(u)</math>, we need to make <math>u</math> as big or as small as possible, depending on the sign of <math>\phi(x,\lambda,t)</math>, specifically:

			: <math>u(t) = \begin{cases} b, & \phi(x,\lambda,t)<0 \\ ?, & \phi(x,\lambda,t)=0 \\ a, & \phi(x,\lambda,t)>0.\end{cases}</math>

			If <math>\phi</math> is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a [[bang-bang control]] that switches from <math>b</math> to <math>a</math> at times when <math>\phi</math> switches from negative to positive.

			The case when <math>\phi</math> remains at zero for a finite length of time <math>t_1\le t\le t_2</math> is called the '''singular control''' case. Between <math>t_1</math> and <math>t_2</math> the maximization of the Hamiltonian with respect to u gives us no useful information and the solution in that time interval is going to have to be found from other considerations. (One approach would be to repeatedly differentiate <math>\partial H/\partial u</math> with respect to time until the control u again explicitly appears, which is guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between <math>t_1</math> and <math>t_2</math> the control <math>u</math> is determined by the requirement that the singularity condition continues to hold. The resulting so-called singular arc will be optimal if it satisfies the '''Kelley condition''':

			:<math>(-1)^k \frac{\partial}{\partial u} \left[ {\left( \frac{d}{dt} \right)}^{2k} H_u \right] \ge 0 ,\, k=0,1,\cdots</math>

			.<ref>Bryson, Ho: Applied Optimal Control, Page 246</ref> This condition is also called the generalized [[Legendre-Clebsch condition]]).

			The term '''bang-singular control''' refers to a control that has a bang-bang portion as well as a singular portion.

			==References==

			{{reflist}}

			{{DEFAULTSORT:Singular Control}}
			[[Category:Control theory]]

210.212.179.137 at 07:43, 1 February 2011

2011-02-01T07:43:56Z

New page

Hi there. My title is Sophia Meagher even though it is not the name on my beginning certification. To climb is some thing I truly appreciate performing. Alaska is exactly where I've usually been living. My working day occupation is an information officer but I've currently applied for another one.<br><br>My web page; telephone psychic ([http://appin.co.kr/board_Zqtv22/688025 appin.co.kr])