Fuller's problem - Revision history

ClemensZeile: /* Variants */

2018-01-08T22:01:27Z

Variants

← Older revision		Revision as of 22:01, 8 January 2018
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	== Variants ==		== Variants ==

	* ~~dropped~~ terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value problem]],		* Dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value problem]],
	* ~~several~~ binary controls and dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value multimode problem]],		* Several binary controls and dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value multimode problem]],

	== Source Code ==		== Source Code ==

ClemensZeile: /* Variants */

2018-01-08T22:01:08Z

Variants

← Older revision		Revision as of 22:01, 8 January 2018
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	* dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value problem]],		* dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value problem]],
			* several binary controls and dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value multimode problem]],

	== Source Code ==		== Source Code ==

ClemensZeile at 15:33, 8 January 2018

2018-01-08T15:33:39Z

← Older revision		Revision as of 15:33, 8 January 2018
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	</gallery>		</gallery>
	-->		-->

			== Variants ==

			* dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value problem]],

	== Source Code ==		== Source Code ==

FelixMueller: /* Source Code */

2016-06-28T07:49:38Z

Source Code

← Older revision		Revision as of 07:49, 28 June 2016
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	* [[:Category:Muscod \| Muscod code]] at [[Fuller's Problem (Muscod)]]		* [[:Category:Muscod \| Muscod code]] at [[Fuller's Problem (Muscod)]]
	* [[:Category:optimica \| optimica]] at [[Fuller's Problem (optimica)]]		* [[:Category:JModelica \| JModelica code]] at [[Fuller's Problem (JModelica)]]
	* [[:Category:JModelica \| JModelica]] at [[Fuller's Problem (JModelica)]]

	== Miscellaneous and further reading ==		== Miscellaneous and further reading ==

FelixMueller: /* Reference Solutions */

2016-04-17T09:47:30Z

Reference Solutions

← Older revision		Revision as of 09:47, 17 April 2016
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	== Reference Solutions ==		== Reference Solutions ==

	===Solutions obtained with ~~optimica~~===		===Solutions obtained with jModelica===

	The solution found for the relaxed Fuller's problem with ~~optimica~~ using the solver Ipopt (with the linear solver MA27) is obtained with 12 iterations and the objective is 1.5296058259296967e-05.		The solution found for the relaxed Fuller's problem with jModelica using the solver Ipopt (with the linear solver MA27) is obtained with 12 iterations and the objective is 1.5296058259296967e-05.
	[[File:Fullerspng.png\|left\|200px\|thumb\|alt=a graph with the optimal solution of the Fuller's Problem with Optimica and Ipopt\|Solution of the Fuller's Problem with Optimica and Ipopt]]		[[File:Fullerspng.png\|left\|200px\|thumb\|alt=a graph with the optimal solution of the Fuller's Problem with Optimica and Ipopt\|Solution of the Fuller's Problem with Optimica and Ipopt]]
	<br />		<br />

MadeleineSchroeter at 12:46, 15 March 2016

2016-03-15T12:46:50Z

← Older revision		Revision as of 12:46, 15 March 2016
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	* [[:Category:Muscod \| Muscod code]] at [[Fuller's Problem (Muscod)]]		* [[:Category:Muscod \| Muscod code]] at [[Fuller's Problem (Muscod)]]
	* [[:Category:optimica \| optimica]] at [[Fuller's Problem (optimica)]]		* [[:Category:optimica \| optimica]] at [[Fuller's Problem (optimica)]]
			* [[:Category:JModelica \| JModelica]] at [[Fuller's Problem (JModelica)]]

	== Miscellaneous and further reading ==		== Miscellaneous and further reading ==

FelixMueller: /* Mathematical formulation */

2016-02-17T13:19:43Z

Mathematical formulation

← Older revision		Revision as of 13:19, 17 February 2016
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	<math>		<math>
	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0 & = & x_1, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
	& \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]		& \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]

FelixMueller: /* Mathematical formulation */

2016-02-17T13:19:25Z

Mathematical formulation

← Older revision		Revision as of 13:19, 17 February 2016
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	<math>		<math>
	\begin{array}{llcl}		\begin{array}{llcl}
	\displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 \; \mathrm{d} t \\[1.5ex]		\displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 \; \mathrm{d} t \\[1.5ex]
	\mbox{s.t.} & \dot{x}_0~~(t)~~ & = & x_1~~(t)~~, \\		\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
	& \dot{x}_1~~(t)~~ & = & 1 - 2 \; w~~(t)~~, \\[1.5ex]		& \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
	& x(0) &=& x_S, \\		& x(0) &=& x_S, \\
	& x(t_f) &=& x_T, \\		& x(t_f) &=& x_T, \\

SebastianSager: Moved C to Muscod

2016-01-28T09:18:30Z

Moved C to Muscod

← Older revision		Revision as of 09:18, 28 January 2016
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	== Source Code ==		== Source Code ==

	* [[:Category:C \| C code]] at [[Fuller's Problem (C)]]		* [[:Category:Muscod \| Muscod code]] at [[Fuller's Problem (Muscod)]]
	* [[:Category:optimica \| optimica]] at [[Fuller's Problem (optimica)]]		* [[:Category:optimica \| optimica]] at [[Fuller's Problem (optimica)]]

JonasSchulze: Text replacement - "\(.*)\<\/bibref\>" to ""

2015-12-30T20:32:09Z

Text replacement - "\<bibref\>(.*)\<\/bibref\>" to "<bib id="$1" />"

← Older revision		Revision as of 20:32, 30 December 2015
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	}}		}}

	The first control problem with an optimal [[:Category:Chattering\|chattering]] solution was given by <~~bibref>~~Fuller1963</~~bibref~~>. An optimal trajectory does exist for all initial and terminal values in a vicinity of the origin. As Fuller showed, this optimal trajectory contains a bang-bang control function that switches infinitely often.		The first control problem with an optimal [[:Category:Chattering\|chattering]] solution was given by <bib id="Fuller1963" />. An optimal trajectory does exist for all initial and terminal values in a vicinity of the origin. As Fuller showed, this optimal trajectory contains a bang-bang control function that switches infinitely often.

	The mathematical equations form a small-scale [[:Category:ODE model\|ODE model]]. The interior point equality conditions fix initial and terminal values of the differential states.		The mathematical equations form a small-scale [[:Category:ODE model\|ODE model]]. The interior point equality conditions fix initial and terminal values of the differential states.
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	== Miscellaneous and further reading ==		== Miscellaneous and further reading ==

	An extensive analytical investigation of this problem and a discussion of the ubiquity of Fuller's problem can be found in <~~bibref>~~Zelikin1994</~~bibref~~>, a recent investigation of chattering controls in relay feedback systems in <~~bibref>~~Johansson2002</~~bibref~~>.		An extensive analytical investigation of this problem and a discussion of the ubiquity of Fuller's problem can be found in <bib id="Zelikin1994" />, a recent investigation of chattering controls in relay feedback systems in <bib id="Johansson2002" />.

	== References ==		== References ==