LV Shared Resource: Difference between revisions

LV Shared Resource
State dimension:	1
Differential states:	3
Discrete control functions:	1

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Latest revision as of 13:43, 28 November 2025

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 ODE model.

Mathematical formulation

The optimal control problem is given by

$\begin{array}{llclr} \min_{u} & \int_{0}^{t_{f}} & (x_{0} (t) - 1.5)^{2} + (x_{1} (t) - 1)^{2} + (x_{2} (t) - 1)^{2} d t \\ s.t. & {\dot{x}}_{0} (t) & = & x_{0} (t) - x_{0} (t) x_{1} (t) - x_{0} (t) x_{2} (t), \\ {\dot{x}}_{1} (t) & = & - x_{1} (t) + x_{0} (t) x_{1} (t) - c_{1} x_{1} (t) u (t), \\ {\dot{x}}_{2} (t) & = & - x_{2} (t) + α x_{0} (t) x_{2} (t) - c_{2} x_{2} (t) u (t), \\ x (0) & = & x_{0}, \\ u (t) & \in & [0, 1], \\ α & > & 1 . \end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_{0}, t_{f}] & = & [0, 40], \\ (c_{1}, c_{2}) & = & (0.1, 0.4), \\ x_{0} & = & (1.5, 0.5, 1) or (1.5, 1, 0.5), \\ α & = & 1.2 . \end{array}$

Reference Solutions

Reference solution plots
Local optimum a direct approach for start values $x_{0} = (1.5, 0.5, 1)$ .
Local optimum a direct approach for start values $x_{0} = (1.5, 1, 0.5)$ .

@@ Line 2: / Line 2: @@
 |nd        = 1
 |nx        = 3
-|nw        = 5
+|nw        = 1
-|nre       = 3
 }}
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 == 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.
+<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>.
-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>.
+  Image:LV_Shared_init_2.png| Local optimum a direct approach for start values <math>x_0 = (1.5, 1, 0.5)</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>
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 [[Category:ODE model]]
 [[Category:Tracking objective]]
-[[Category:Chattering]]
 [[Category:Sensitivity-seeking arcs]]
 [[Category:Population dynamics]]