Lotka Experimental Design - Revision history

RobertLampel: /* Mathematical formulation */

2025-10-15T08:26:09Z

Mathematical formulation

← Older revision		Revision as of 08:26, 15 October 2025
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	\quad x(0) &=& (0.5, 0.7), \\		\quad x(0) &=& (0.5, 0.7), \\
	\quad G(0) &=& F(0) = 0, \\		\quad G(0) &=& F(0) = 0, \\
	\quad z^1(0) &=& z^2(0) = 0, \\ [1.5ex]		\quad z^1(0) &=& z^2(0) = 0, \\[1.5ex]
	\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\		\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\
	\quad 0 & \le & M - z(t_f)		\quad 0 & \le & M - z(t_f)
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	</p>		</p>

	with		with
	<math>f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t)~~</math>~~,
	~~<math>~~f_{x12}(\cdot) = - p_2 x_1(t)~~</math>~~,		<math>
	~~<math>~~f_{x21}(\cdot) = p_4 x_2(t)~~</math>~~,		\begin{align}
	~~<math>~~f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t)~~</math>~~, ~~and~~		f_{x11}(\cdot) &= \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t), \\
	~~<math>~~f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t)~~</math>~~,		f_{x12}(\cdot) &= - p_2 x_1(t), \\
	~~<math>~~f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)</math>.		f_{x21}(\cdot) &= p_4 x_2(t), \\
			f_{x22}(\cdot) &= -p_3 + p_4 x_1(t) - p_6 u(t), \\
			f_{p12}(\cdot) &= \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t), \text{ and}\\
			f_{p24}(\cdot) &= \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)
			\end{align}
			</math>.

	Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.		Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.

FelixMueller at 08:31, 27 July 2016

2016-07-27T08:31:11Z

← Older revision		Revision as of 08:31, 27 July 2016
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	\|nx = 11		\|nx = 11
	\|nw = 2		\|nw = 2
			\|nc = 4
	\|nre = 11		\|nre = 11
	}}		}}

FelixMueller: /* Source Code */

2016-01-19T15:48:09Z

Source Code

← Older revision		Revision as of 15:48, 19 January 2016
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	* [[:Category:AMPL \| AMPL]] at [[Lotka Experimental Design (AMPL)]]		* [[:Category:AMPL \| AMPL]] at [[Lotka Experimental Design (AMPL)]]
			* [[:Category: VPLAN \| VPLAN code]] at [[Lotka Experimental Design (VPLAN)]]

	== Variants ==		== Variants ==

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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	There are several alternative formulations and variants of the above problem, in particular		There are several alternative formulations and variants of the above problem, in particular

	* a prescribed time grid for the control function <~~bibref>~~Sager2006</~~bibref~~>, see also [[Lotka Experimental Design (AMPL)]],		* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Experimental Design (AMPL)]],
	* no fishing, i.e., <math>u \equiv 0</math>,		* no fishing, i.e., <math>u \equiv 0</math>,
	* different fishing control functions for the two species,		* different fishing control functions for the two species,
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <~~bibref>~~Sager2006</~~bibref~~> and revisited in his PhD thesis <~~bibref>~~Sager2005</~~bibref~~>. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <~~bibref>~~Sager2011d</~~bibref~~>.		The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

	== References ==		== References ==

JonasSchulze: Text replacement - "" to ""

2015-12-30T20:28:26Z

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

← Older revision		Revision as of 20:28, 30 December 2015
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	== 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 -->

SebastianSager: correct matrix F

2012-10-19T10:11:50Z

correct matrix F

← Older revision		Revision as of 10:11, 19 October 2012
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	\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\		\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\
	\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\		\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\
	\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{12}(t)^2, \\		\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{21}(t)^2, \\
	\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{12}(t) G_{22}(t), \\		\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{21}(t) G_{22}(t), \\
	\quad \dot{F_{22}}(t) & = & w^1(t) G_{21}(t)^2 + w^2(t) G_{22}(t)^2, \\		\quad \dot{F_{22}}(t) & = & w^1(t) G_{12}(t)^2 + w^2(t) G_{22}(t)^2, \\
	\quad \dot{z^1}(t) & = & w^1(t), \\		\quad \dot{z^1}(t) & = & w^1(t), \\
	\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]		\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]

SebastianSager at 08:39, 31 August 2012

2012-08-31T08:39:00Z

← Older revision		Revision as of 08:39, 31 August 2012
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	\|nx = 11		\|nx = 11
	\|nw = 2		\|nw = 2
	\|nre = 12		\|nre = 11
	}}		}}

SebastianSager at 08:38, 31 August 2012

2012-08-31T08:38:50Z

← Older revision		Revision as of 08:38, 31 August 2012
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	{{Dimensions		{{Dimensions
	\|nd = 1		\|nd = 1
	\|nx = 12		\|nx = 11
	\|nw = 2		\|nw = 2
	\|nre = 12		\|nre = 12

SebastianSager: plots uploaded

2012-08-31T08:38:25Z

plots uploaded

← Older revision		Revision as of 08:38, 31 August 2012
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	== Reference Solutions ==		== Reference Solutions ==

	~~bla~~		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~lotkaRelaxedControls~~.png\| ~~Optimal relaxed control determined by an indirect approach~~ and ~~by a direct approach on different~~ control ~~discretization grids~~.		Image:vplotkaUStates.png\| States and fishing control.
	Image:~~lotkaindirektStates~~.png\| ~~Differential states and corresponding adjoint variables in the indirect approach~~.		Image:vplotkaUSensitivities.png\| Sensitivities G().
	Image:~~lotka2Switches~~.png\| ~~Control and differential states with only two switches~~.		Image:vplotkaUMeas1.png\| Sampling function for first state.
	Image:~~lotka56Switches~~.png\| ~~Control and differential states with 56 switches~~.		Image:vplotkaUMeas2.png\| Sampling function for second state.
	</gallery>		</gallery>


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

SebastianSager: Model description is complete

2012-08-31T08:13:25Z

Model description is complete

← Older revision		Revision as of 08:13, 31 August 2012
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	</math>		</math>
	</p>		</p>
	where <math>u(\cdot)</math> is a fishing control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 12</math> are fixed. We are interested in how to fish and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w^1(\cdot)</math> and <math>w^2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>~~\Ge~~(\cdot)</math> are identical. The experimental design problem then reads		where <math>u(\cdot)</math> is a fishing control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 12</math> are fixed. We are interested in how to fish and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w^1(\cdot)</math> and <math>w^2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical. The experimental design problem then reads


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

	<p>		<p>
	<math>		<math>
	\begin{array}{~~llcl~~}		\begin{array}{lll}
	\displaystyle \min_{x, w} & ~~x_2~~(t_f) \\~~[1.5ex]~~		\displaystyle \min_{x,G,F,z^1,z^2,u,w^1,w^2} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
	\~~mbox~~{~~s.t.~~} & \dot{x}_0(t) & = & ~~x_0~~(t) - ~~x_0~~(t) x_1(t) - \; ~~c_0 x_0~~(t) \; w(t), \\		\text{subject to} \\
	& \dot{x}_1(t) & = & ~~- x_1~~(t) + ~~x_0~~(t) ~~x_1~~(t) - \; ~~c_1 x_1~~(t) \; w(t), \\		\quad \dot{x_1}(t) & = & p_1 \; x_1(t) - p_2 x_1(t) x_2(t) - p_5 u(t) x_1(t),\\
	& \dot{x}_2(t) & = & (~~x_0~~(t) - 1)^2 + (~~x_1~~(t) - 1)^2, \\[1.5ex]		\quad \dot{x_2}(t) & = & - p_3 \; x_2(t) + p_4 x_1(t) x_2(t) - p_6 u(t) x_2(t),\\
	& x(0) &=& (0.5, 0.7, 0)^T, \\		\quad \dot{G_{11}}(t) & = & f_{x11}(\cdot) \; G_{11}(t) + f_{x12}(\cdot) \; G_{21}(t) + f_{p12}(\cdot), \\
	~~& w~~(t) &\in& \{0, 1\}.		\quad \dot{G_{12}}(t) & = & f_{x11}(\cdot) \; G_{12}(t) + f_{x12}(\cdot) \; G_{22}(t), \\
	\end{array}		\quad \dot{G_{21}}(t) & = & f_{x21}(\cdot) \; G_{11}(t) + f_{x22}(\cdot) \; G_{21}(t), \\
			\quad \dot{G_{22}}(t) & = & f_{x21}(\cdot) \; G_{12}(t) + f_{x22}(\cdot) \; G_{22}(t) + f_{p24}(\cdot), \\
			\quad \dot{F_{11}}(t) & = & w^1(t) G_{11}(t)^2 + w^2(t) G_{12}(t)^2, \\
			\quad \dot{F_{12}}(t) & = & w^1(t) G_{11}(t) G_{12}(t) + w^2(t) G_{12}(t) G_{22}(t), \\
			\quad \dot{F_{22}}(t) & = & w^1(t) G_{21}(t)^2 + w^2(t) G_{22}(t)^2, \\
			\quad \dot{z^1}(t) & = & w^1(t), \\
			\quad \dot{z^2}(t) & = & w^2(t), \\[1.5ex]
			\quad x(0) &=& (0.5, 0.7), \\
			\quad G(0) &=& F(0) = 0, \\
			\quad z^1(0) &=& z^2(0) = 0, \\ [1.5ex]
			\quad u(t) & \in & \mathcal{U}, \; w^1(t) \in \mathcal{W}, \; w^2(t) \in \mathcal{W}, \\
			\quad 0 & \le & M - z(t_f)
			\end{array}
	</math>		</math>
	</p>		</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>~~. The decision~~, ~~whether the fishing fleet is actually fishing at time~~ <math>t</math> ~~is denoted by~~ <math>w(t)</math>.		with
			<math>f_{x11}(\cdot) = \partial f_1(\cdot) / \partial x_1 = p_1 - p_2 x_2(t) - p_5 u(t)</math>,
			<math>f_{x12}(\cdot) = - p_2 x_1(t)</math>,
			<math>f_{x21}(\cdot) = p_4 x_2(t)</math>,
			<math>f_{x22}(\cdot) = -p_3 + p_4 x_1(t) - p_6 u(t)</math>, and
			<math>f_{p12}(\cdot) = \partial f_1(\cdot) / \partial p_2 = -x_1(t) x_2(t)</math>,
			<math>f_{p24}(\cdot) = \partial f_2(\cdot) / \partial p_4 = x_1(t) x_2(t)</math>.

			Note that the state <math>F_{21}(\cdot) = F_{12}(\cdot)</math> has been left out for reasons of symmetry.

	== Parameters ==		== Parameters ==

	~~These fixed values are used within the model.~~		We use <math>t_f=12</math>, <math>p_1 = p_2 = p_3 = p_4 = 1</math>, and <math>p_5 = 0.4</math>, <math>p_6 = 0.2</math>. The upper bound on the measurement time intervals is chosen as <math>M=4</math>.

	<math>
	~~\begin{array}{rcl}~~
	~~[t_0,~~ t_f~~] &~~=~~& [0,~~ 12],\\
	~~(c_0~~, ~~c_1) &~~=~~& (~~0.4, 0.2).
	~~\end{array}~~
	</math>

	== Reference Solutions ==		== 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 [http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle Pontryagins maximum principle]. The optimal solution contains a singular arc, as can be seen in the plot of the optimal control. The two differential states and corresponding adjoint variables in the indirect approach are also displayed. A different approach to solving the relaxed problem is by using a direct method such as collocation or Bock's direct multiple shooting method. Optimal solutions for different control discretizations are also plotted in the leftmost figure.		bla

	The optimal objective value of this relaxed problem is <math>x_2(t_f) = 1.34408</math>. As follows from MIOC theory<bibref>Sager2008</bibref> this is the best lower bound on the optimal value of the original problem with the integer restriction on the control function. In other words, this objective value can be approximated arbitrarily close, if the control only switches often enough between 0 and 1. As no optimal solution exists, two suboptimal ones are shown, one with only two switches and an objective function value of <math>x_2(t_f) = 1.38276</math>, and one with 56 switches and <math>x_2(t_f) = 1.34416</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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	* a prescribed time grid for the control function <bibref>Sager2006</bibref>, see also [[Lotka Experimental Design (AMPL)]],		* a prescribed time grid for the control function <bibref>Sager2006</bibref>, see also [[Lotka Experimental Design (AMPL)]],
			* no fishing, i.e., <math>u \equiv 0</math>,
	* different fishing control functions for the two species,		* different fishing control functions for the two species,
	* different parameters and start values.		* different parameters and start values.