Bang-bang approximation of a traveling wave - Revision history

FelixMueller at 15:02, 27 January 2016

2016-01-27T15:02:45Z

← Older revision		Revision as of 15:02, 27 January 2016
Line 1:		Line 1:
			{{Dimensions
			\|nd = 2
			\|nx = 1
			\|nw = 1
			\|nc = 2
			}}

	The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints		The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
	and was introduced in <bib id="Hante2009" />.		and was introduced in <bib id="Hante2009" />.

2015-12-30T20:32:10Z

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

← Older revision		Revision as of 20:32, 30 December 2015
Line 1:		Line 1:
	The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints		The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
	and was introduced in <~~bibref>~~Hante2009</~~bibref~~>.		and was introduced in <bib id="Hante2009" />.

	The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling		The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling

2015-12-30T20:28:25Z

Text replacement - "<bibreferences/>" to "<biblist />"

← Older revision		Revision as of 20:28, 30 December 2015
Line 47:		Line 47:

	== References ==		== References ==
	<~~bibreferences~~/>		<biblist />

	<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->		<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

2010-11-21T15:02:14Z

← Older revision		Revision as of 15:02, 21 November 2010
Line 28:		Line 28:
	Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the		Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the
	characteristic equations.		characteristic equations.
			Systems biology

	== Reference solution ==		== Reference solution ==
Line 53:		Line 53:
	[[Category:PDE model]]		[[Category:PDE model]]
	[[Category:Transport]]		[[Category:Transport]]
			[[Category:Tracking objective]]

2010-08-24T09:27:35Z

← Older revision		Revision as of 09:27, 24 August 2010
Line 41:		Line 41:

	<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">		<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">
	Image:sinPGsws.png\| Solution obtained using a projected gradient method based on switching time sensitivities.		Image:sinPGsws.png\| Solution <math>q^*(t)</math> obtained using a projected gradient method based on switching time sensitivities.
	Image:sinFinalTime.png\| Corresponding ~~final time plot~~ of the differential states (blue) and the wave (red).		Image:sinFinalTime.png\| Corresponding plots of the differential states (blue) and the wave (red) at <math>t=1</math>.
	</gallery>		</gallery>

2010-08-24T09:24:15Z

Added figures

← Older revision		Revision as of 09:24, 24 August 2010
Line 39:		Line 39:

	where <math>\chi_{[a,b]}(t)</math> denotes the indicator function of the interval <math>[a,b]</math>.		where <math>\chi_{[a,b]}(t)</math> denotes the indicator function of the interval <math>[a,b]</math>.

			<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">
			Image:sinPGsws.png\| Solution obtained using a projected gradient method based on switching time sensitivities.
			Image:sinFinalTime.png\| Corresponding final time plot of the differential states (blue) and the wave (red).
			</gallery>

2010-08-16T12:47:44Z

← Older revision		Revision as of 12:47, 16 August 2010
Line 25:		Line 25:

	is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and		is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
	<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of q(\cdot) over the interval <math>[0,1]</math>.		<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of <math>q(\cdot)</math> over the interval <math>[0,1]</math>.
	Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the		Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the
	characteristic equations.		characteristic equations.

2010-08-16T12:31:15Z

typo

← Older revision		Revision as of 12:31, 16 August 2010
Line 30:		Line 30:


	== Reference ~~Solution~~ ==		== Reference solution ==

	For <math>c=0.0075</math> the best known solution is given by		For <math>c=0.0075</math> the best known solution is given by

2010-08-16T12:29:12Z

Added reference

← Older revision		Revision as of 12:29, 16 August 2010
Line 1:		Line 1:
	The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints.		The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
	~~<!--~~ and was introduced in <bibref>Hante2009</bibref>. ~~-->~~		and was introduced in <bibref>Hante2009</bibref>.

	The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling		The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling

2010-08-16T12:20:45Z

Added reference solution

← Older revision		Revision as of 12:20, 16 August 2010
Line 4:		Line 4:
	The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling		The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling
	wave such that the <math>L^2</math>-distance between the traveling wave and the resulting flow is minimized.		wave such that the <math>L^2</math>-distance between the traveling wave and the resulting flow is minimized.


	== Mathematical formulation ==		== Mathematical formulation ==
Line 27:		Line 28:
	Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the		Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the
	characteristic equations.		characteristic equations.


			== Reference Solution ==

			For <math>c=0.0075</math> the best known solution is given by

			<math>
			q^*(t)=\chi_{[0,0.2]}(t)+\chi_{[0.4,0.6]}(t)+\chi_{[0.8,1]}(t),\quad 0\leq t\leq 1
			</math>

			where <math>\chi_{[a,b]}(t)</math> denotes the indicator function of the interval <math>[a,b]</math>.


	== References ==		== References ==