Fuller's initial value problem - Revision history

ClemensZeile: /* Parameters */

2018-01-08T22:26:53Z

Parameters

← Older revision		Revision as of 22:26, 8 January 2018
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	== Parameters ==		== Parameters ==

	We use <math>x_S = x_T = (0.01, 0)^T</math>.		We use <math>x_S = x_T = (0.01, 0)^T</math> and <math>(t_0,t_f) = (0,1)</math>.

	== Reference Solutions ==		== Reference Solutions ==

ClemensZeile: /* Mathematical formulation */

2018-01-08T22:26:10Z

Mathematical formulation

← Older revision		Revision as of 22:26, 8 January 2018
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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 + (x(t_f)-x_T)^2 \\[1.5ex]		\displaystyle \min_{x, w} & \int_{t_0}^{t_f} x_0^2 \; &\mathrm{d} t& + (x(t_f)-x_T)^2 \\[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]

ClemensZeile at 16:31, 8 January 2018

2018-01-08T16:31:28Z

← Older revision		Revision as of 16:31, 8 January 2018
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	</math>		</math>
	</p>		</p>


	== Parameters ==		== Parameters ==

	We use <math>x_S = x_T = (0.01, 0)^T</math>.		We use <math>x_S = x_T = (0.01, 0)^T</math>.


	== Reference Solutions ==		== Reference Solutions ==

ClemensZeile: /* Reference Solutions */

2018-01-08T16:28:54Z

Reference Solutions

← Older revision		Revision as of 16:28, 8 January 2018
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
	Image:FullerRelaxed 6000 40 1.png\| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=150</math>.		Image:FullerRelaxed 6000 40 1.png\| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=150</math>.
	Image:~~MmlotkaCIA 12000 30~~ 1.png\| Optimal binary controls ~~and states~~ determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=~~12000~~, \, n_u=~~400~~</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:FullerRelaxed 6000 40 2.png\| Optimal relaxed controls.
			Image:FullerCIA 6000 40 1.png\| Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
			Image:FullerCIA 6000 40 2.png\| Optimal binary controls.
	</gallery>		</gallery>

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	[[Category:Chattering]]		[[Category:Chattering]]
	[[Category:Sensitivity-seeking arcs]]		[[Category:Sensitivity-seeking arcs]]
	~~[[Category:Population dynamics]]~~

ClemensZeile: /* Reference Solutions */

2018-01-08T16:22:32Z

Reference Solutions

← Older revision		Revision as of 16:22, 8 January 2018
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	If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.		If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

	The optimal objective value of the relaxed problem with <math> n_t=~~12000~~, \, n_u=~~400~~ </math> is <math>~~x_2(t_f) =~~1.~~82875272~~</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>~~x_2(t_f) =1~~.~~82878681~~</math>.		The optimal objective value of the relaxed problem with <math> n_t=6000, \, n_u=150 </math> is <math>1.45412214e-05</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>2.40273813e-05</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
	Image:~~MmlotkaRelaxed_12000_30_1~~.png\| Optimal relaxed ~~controls and~~ states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=~~12000~~, \, n_u=~~400~~</math>.		Image:FullerRelaxed 6000 40 1.png\| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=6000, \, n_u=150</math>.
	Image:MmlotkaCIA 12000 30 1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:MmlotkaCIA 12000 30 1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>

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2018-01-08T15:45:07Z

Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 |nre = 2 }}This site describ..."

New page

{{Dimensions
|nd = 1
|nx = 2
|nw = 1
|nre = 2
}}This site describes a Fuller's problem variant with no terminal constraints and additional Mayer term for penalizing deviation from given reference values.

== Mathematical formulation ==

For <math>t \in [t_0, t_f]</math> almost everywhere the mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llcl}
\displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t + (x(t_f)-x_T)^2 \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 & = & x_1, \\
& \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
& x(0) &=& x_S, \\
& w(t) &\in& \{0, 1\}.
\end{array}
</math>
</p>

== Parameters ==

We use <math>x_S = x_T = (0.01, 0)^T</math>.

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with <math> n_t=12000, \, n_u=400 </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:MmlotkaRelaxed_12000_30_1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=400</math>.
Image:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>


[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]