Lotka Volterra multi-arcs problem - Revision history

ClemensZeile at 13:17, 21 December 2017

2017-12-21T13:17:23Z

← Older revision		Revision as of 13:17, 21 December 2017
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	\begin{array}{rcl}		\begin{array}{rcl}
	[t_0, t_f] &=& [0, 18],\\		[t_0, t_f] &=& [0, 18],\\
	(c_0, c_1,) &=& (0.4, 0.2),\\		(c_0, c_1) &=& (0.4, 0.2),\\
	p_{0}(t) &=& \left\{ \begin{array}{ll}		p_{0}(t) &=& \left\{ \begin{array}{ll}
	1 \quad \quad &\text{for } \ t\in [0,7.2],\\		1 \quad \quad &\text{for } \ t\in [0,7.2],\\

ClemensZeile at 13:15, 21 December 2017

2017-12-21T13:15:26Z

← Older revision		Revision as of 13:15, 21 December 2017
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	1 \quad \quad &\text{for } \ t\in [12,18].		1 \quad \quad &\text{for } \ t\in [12,18].
	\end{array} \right. \\		\end{array} \right. \\
	p_1(t) &=& p_0(t).		p_1(t) &\equiv& p_0(t).
	\end{array}		\end{array}
	</math>		</math>
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<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=18000, \, n_u=300</math>.		Image:LotkaRelaxed 18000 60 1.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=18000, \, n_u=300</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=18000, \, n_u=300</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.		Image:LotkaCIA 18000 60 1.png\| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=18000, \, n_u=300</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>

ClemensZeile at 13:10, 21 December 2017

2017-12-21T13:10:02Z

← Older revision		Revision as of 13:10, 21 December 2017
Line 36:		Line 36:
	[t_0, t_f] &=& [0, 18],\\		[t_0, t_f] &=& [0, 18],\\
	(c_0, c_1,) &=& (0.4, 0.2),\\		(c_0, c_1,) &=& (0.4, 0.2),\\
	p_{0}(t) &=& \left\{\begin{array}[ll]1 & \text{for } t\in[0, 7.2] \end{array} \right. \\		p_{0}(t) &=& \left\{ \begin{array}{ll}
			1 \quad \quad &\text{for } \ t\in [0,7.2],\\
			0.9 \quad \quad &\text{for } \ t\in [7.2,12],\\
			1 \quad \quad &\text{for } \ t\in [12,18].
			\end{array} \right. \\
	p_1(t) &=& p_0(t).		p_1(t) &=& p_0(t).
	\end{array}		\end{array}
Line 45:		Line 49:
	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=18000, \, n_u=300 </math> is <math>x_2(t_f) =1.41842699</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.4237735 </math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<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:MmlotkaRelaxed_12000_30_1.png\| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=18000, \, n_u=300</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=18000, \, n_u=300</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
	</gallery>		</gallery>

ClemensZeile: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 |nre = 3 }}This site describ..."

2017-12-21T13:00:21Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 |nre = 3 }}This site describ..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
|nre = 3
}}This site describes a Lotka Volterra variant with three singular arcs instead of only one in the standard variant.

== Mathematical formulation ==

The mixed-integer optimal control problem is given by

<p>
<math>
\begin{array}{llclr}
\displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex]
\mbox{s.t.}
& \dot{x}_0 & = & x_0 - x_0 x_1 - \; c_0\; x_0 \; w, \\
& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; c_1\; x_1 \; w, \\
& \dot{x}_2 & = & (x_0 - p_0(t))^2 + (x_1 - p_1(t))^2, \\[1.5ex]
& x(0) &=& (0.5, 0.7, 0)^T, \\
& w(t) &\in& \{0, 1\},
\end{array}
</math>
</p>

Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between three different fishing options.

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 18],\\
(c_0, c_1,) &=& (0.4, 0.2),\\
p_{0}(t) &=& \left\{\begin{array}[ll]1 & \text{for } t\in[0, 7.2] \end{array} \right. \\
p_1(t) &=& p_0(t).
\end{array}
</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]]