LV Shared Resource - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T13:43:51Z

Reference Solutions

← Older revision		Revision as of 13:43, 28 November 2025
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	== Reference Solutions ==		== Reference Solutions ==

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="2">		<gallery caption="Reference solution plots" widths="400px" heights="240px" perrow="2">
	Image:LV_Shared_init_1.png\| Local optimum a direct approach for start values <math>x_0 = (1.5, 0.5, 1)</math>.		Image:LV_Shared_init_1.png\| Local optimum a direct approach for start values <math>x_0 = (1.5, 0.5, 1)</math>.
	Image:LV_Shared_init_2.png\| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</math>.		Image:LV_Shared_init_2.png\| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</math>.

RobertLampel at 07:27, 25 August 2025

2025-08-25T07:27:22Z

← Older revision		Revision as of 07:27, 25 August 2025
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	\|nd = 1		\|nd = 1
	\|nx = 3		\|nx = 3
	\|nw = 5		\|nw = 1
	~~\|nre = 3~~
	}}		}}

RobertLampel: /* Reference Solutions */

2025-08-25T07:16:53Z

Reference Solutions

← Older revision		Revision as of 07:16, 25 August 2025
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	== Reference Solutions ==		== Reference Solutions ==

	If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, 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=150 </math> is <math>x_2(t_f) =0.345768563</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =0.348617982</math>.

	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
	Image:~~Lotka_abs_fish_relaxed_12000_80~~.~~pdf~~\| ~~Optimal relaxed controls and states determined by an~~ direct approach ~~with ampl_mintoc (Radau collocation) and~~ <math>~~n_t~~=~~12000~~, \, ~~n_u=150~~</math>.		Image:LV_Shared_init_1.png\| Local optimum a direct approach for start values <math>x_0 = (1.5, 0.5, 1)</math>.
	Image:~~Lotka_abs_fish_CIA_states_12000_80~~.~~pdf~~\| ~~Differential states determined by an~~ direct approach ~~(Radau collocation) with ampl_mintoc and~~ <math>~~n_t~~=~~12000~~, \, ~~n_u=150</math>~~. ~~The relaxed controls were approximated by Combinatorial Integral Approximation.~~		Image:LV_Shared_init_2.png\| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</math>.
	~~Image:Lotka_abs_fish_CIA_controls_12000_80.pdf\| Binary control determined by an direct approach (Radau collocation~~) ~~with ampl_mintoc and <math>n_t=12000, \, n_u=150~~</math>~~. The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering~~.
	</gallery>		</gallery>








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

RobertLampel: /* Mathematical formulation */

2025-08-25T07:01:47Z

Mathematical formulation

← Older revision		Revision as of 07:01, 25 August 2025
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	\displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1.5)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 \ dt \\[1.5ex]		\displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1.5)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 \ dt \\[1.5ex]
	\mbox{s.t.}		\mbox{s.t.}
	& \dot{x}_0 & = & x_0(t) - x_0(t) x_1(t) - x_0(t) x_2(t), \\		& \dot{x}_0(t) & = & x_0(t) - x_0(t) x_1(t) - x_0(t) x_2(t), \\
	& \dot{x}_1 & = & - x_1(t) + x_0(t) x_1(t) - c_1 x_1(t) u(t), \\		& \dot{x}_1(t) & = & - x_1(t) + x_0(t) x_1(t) - c_1 x_1(t) u(t), \\
	& \dot{x}_2 & = & -x_2(t) + \alpha x_0(t) x_2(t) - c_2 x_2(t) u(t), \\[1.5ex]		& \dot{x}_2(t) & = & -x_2(t) + \alpha x_0(t) x_2(t) - c_2 x_2(t) u(t), \\[1.5ex]
	& x(0) &=& x_0, \\		& x(0) &=& x_0, \\
	& u(t) &\in& [0,1], \\		& u(t) &\in& [0,1], \\

RobertLampel: /* Parameters */

2025-08-25T07:01:26Z

Parameters

← Older revision		Revision as of 07:01, 25 August 2025
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	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	[t_0, t_f] &=& [0, 12],\\		[t_0, t_f] &=& [0, 40],\\
	(c_{0,1}, c_{~~1,1~~}) &=& (~~0.2,~~ 0.1),\\		(c_{1}, c_{2}) &=& (0.1, 0.4),\\
	~~(c_{0,2}, c_{1,2}) &=& (~~0.4~~, 0.2~~),\\		x_0 &=& (1.5, 0.5, 1) \text{ or } (1.5, 1, 0.5),\\
	~~(c_{0,3}, c_{1,3})~~ &=& (0.01, 0.1),\\		\alpha &=& 1.2.
	~~(c_~~{~~0,4~~}, ~~c_{~~1~~,4}) &=& (0~~, 0),\\
	~~(c_{0,5}, c_{1,5})~~ &=& ~~(-0.~~1~~, -0~~.2).
	\end{array}		\end{array}
	</math>		</math>

RobertLampel: /* Mathematical formulation */

2025-08-25T06:58:08Z

Mathematical formulation

← Older revision		Revision as of 06:58, 25 August 2025
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	<math>		<math>
	\begin{array}{llclr}		\begin{array}{llclr}
	\displaystyle \min_{u} & \int_0^{t_f} (x_0(t) - 1.5)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 dt \\[1.5ex]		\displaystyle \min_{u} & \int_0^{t_f} && (x_0(t) - 1.5)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 \ dt \\[1.5ex]
	\mbox{s.t.}		\mbox{s.t.}
	& \dot{x}_0 & = & x_0(t) - x_0(t) x_1(t) - x_0(t) x_2(t), \\		& \dot{x}_0 & = & x_0(t) - x_0(t) x_1(t) - x_0(t) x_2(t), \\
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	& \dot{x}_2 & = & -x_2(t) + \alpha x_0(t) x_2(t) - c_2 x_2(t) u(t), \\[1.5ex]		& \dot{x}_2 & = & -x_2(t) + \alpha x_0(t) x_2(t) - c_2 x_2(t) u(t), \\[1.5ex]
	& x(0) &=& x_0, \\		& x(0) &=& x_0, \\
	& u(t) &\in& [0,1],		& u(t) &\in& [0,1], \\
	& \alpha &>& 1.		& \alpha &>& 1.
	\end{array}		\end{array}

RobertLampel: /* Mathematical formulation */

2025-08-25T06:57:22Z

Mathematical formulation

← Older revision		Revision as of 06:57, 25 August 2025
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	== Mathematical formulation ==		== Mathematical formulation ==

	The ~~mixed-integer~~ optimal control problem is given by		The optimal control problem is given by

	<p>		<p>
	<math>		<math>
	\begin{array}{llclr}		\begin{array}{llclr}
	\displaystyle \min_{~~x, w~~} & x_2(~~t_f~~) \\[1.5ex]		\displaystyle \min_{u} & \int_0^{t_f} (x_0(t) - 1.5)^2 + (x_1(t) - 1)^2 + (x_2(t) - 1)^2 dt \\[1.5ex]
	\mbox{s.t.}		\mbox{s.t.}
	& \dot{x}_0 & = & x_0 - x_0 x_1 - ~~\; \sum\limits_{i=1}^{5} c_{0,i}\; w_i~~, \\		& \dot{x}_0 & = & x_0(t) - x_0(t) x_1(t) - x_0(t) x_2(t), \\
	& \dot{x}_1 & = & - x_1 + x_0 x_1 - ~~\; \sum\limits_{i=1}^{5} c_{1,i}\; w_i~~, \\		& \dot{x}_1 & = & - x_1(t) + x_0(t) x_1(t) - c_1 x_1(t) u(t), \\
	& \dot{x}_2 & = & (x_0 - 1)~~^2 +~~ (~~x_1 - 1~~)^2, \\[1.5ex]		& \dot{x}_2 & = & -x_2(t) + \alpha x_0(t) x_2(t) - c_2 x_2(t) u(t), \\[1.5ex]
	& x(0) &=& ~~(0.5, 0.7, 0)^T~~, \\		& x(0) &=& x_0, \\
	& ~~\sum\limits_{i=1}^{5}w_i~~(t) &=& 1, \\		& u(t) &\in& [0,1],
	& ~~w_i(t)~~ &~~\in~~& ~~\{0,~~ 1~~\}, \quad i=1\ldots 5~~.		& \alpha &>& 1.
	\end{array}		\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>. This problem variant allows to choose between five different fishing options.

	== Parameters ==		== Parameters ==

RobertLampel at 06:49, 25 August 2025

2025-08-25T06:49:35Z

← Older revision		Revision as of 06:49, 25 August 2025
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	}}		}}

	This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the [[:Lotka Volterra fishing problem]]. Its dynamics are given via a three-dimensional [[Category:~~OED_model~~]].		This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the [[:Lotka Volterra fishing problem]]. Its dynamics are given via a three-dimensional [[:Category:ODE model\|ODE model]].

	== Mathematical formulation ==		== Mathematical formulation ==

RobertLampel at 06:49, 25 August 2025

2025-08-25T06:49:10Z

← Older revision		Revision as of 06:49, 25 August 2025
Line 6:		Line 6:
	}}		}}

	This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the [[:~~Lotka_Volterra_fishing_problem~~]]. Its dynamics are given via a three-dimensional [[:Category:OED_model]].		This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the [[:Lotka Volterra fishing problem]]. Its dynamics are given via a three-dimensional [[Category:OED_model]].

	== Mathematical formulation ==		== Mathematical formulation ==

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 5 |nre = 3 }} This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the :Lotka_Volterra_fishing_problem. Its dynamics are given via a three-dimensional :Category:OED_model. == Mathematical formulation == The mixed-integer optimal control problem is given by
$\begin{array}{llclr} \displaystyle \min_{x, w} & x_2(t_f) \\[1.5ex] \mbox{s.t.} & \d..."$

2025-08-25T06:48:39Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 5 |nre = 3 }} This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the Lotka_Volterra_fishing_problem. Its dynamics are given via a three-dimensional Category:OED_model. == 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.} & \d..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 5
|nre = 3
}}

This '''Lotka Volterra problem with explicit inclusion of a shared resource''' is a variant of the [[:Lotka_Volterra_fishing_problem]]. Its dynamics are given via a three-dimensional [[:Category:OED_model]].

== 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 - \; \sum\limits_{i=1}^{5} c_{0,i}\; w_i, \\
& \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{5} c_{1,i}\; w_i, \\
& \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]
& x(0) &=& (0.5, 0.7, 0)^T, \\
& \sum\limits_{i=1}^{5}w_i(t) &=& 1, \\
& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 5.
\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 five different fishing options.

== Parameters ==

These fixed values are used within the model.

<math>
\begin{array}{rcl}
[t_0, t_f] &=& [0, 12],\\
(c_{0,1}, c_{1,1}) &=& (0.2, 0.1),\\
(c_{0,2}, c_{1,2}) &=& (0.4, 0.2),\\
(c_{0,3}, c_{1,3}) &=& (0.01, 0.1),\\
(c_{0,4}, c_{1,4}) &=& (0, 0),\\
(c_{0,5}, c_{1,5}) &=& (-0.1, -0.2).
\end{array}
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, 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=150 </math> is <math>x_2(t_f) =0.345768563</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =0.348617982</math>.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:Lotka_abs_fish_relaxed_12000_80.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and <math>n_t=12000, \, n_u=150</math>.
Image:Lotka_abs_fish_CIA_states_12000_80.pdf| Differential states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
Image:Lotka_abs_fish_CIA_controls_12000_80.pdf| Binary control determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering.
</gallery>


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