LV Shared Resource: Difference between revisions

LV Shared Resource
State dimension:	1
Differential states:	3
Discrete control functions:	5
Interior point equalities:	3

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Revision as of 06:57, 25 August 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

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

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_{0}, t_{f}] & = & [0, 12], \\ (c_{0, 1}, c_{1, 1}) & = & (0.2, 0.1), \\ (c_{0, 2}, c_{1, 2}) & = & (0.4, 0.2), \\ (c_{0, 3}, c_{1, 3}) & = & (0.01, 0.1), \\ (c_{0, 4}, c_{1, 4}) & = & (0, 0), \\ (c_{0, 5}, c_{1, 5}) & = & (- 0.1, - 0.2) . \end{array}$

Reference Solutions

If the problem is relaxed, i.e., we demand that $w (t)$ is in the continuous interval $[0, 1]$ 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 $n_{t} = 12000, n_{u} = 150$ is $x_{2} (t_{f}) = 0.345768563$ . The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is $x_{2} (t_{f}) = 0.348617982$ .

Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 12000, n_{u} = 150$ .
Differential states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_{t} = 12000, n_{u} = 150$ . The relaxed controls were approximated by Combinatorial Integral Approximation.
Binary control determined by an direct approach (Radau collocation) with ampl_mintoc and $n_{t} = 12000, n_{u} = 150$ . The relaxed controls were approximated by Combinatorial Integral Approximation. The fishing control shows a lot of chattering.

@@ Line 10: / Line 10: @@
 == Mathematical formulation ==
-The mixed-integer optimal control problem is given by
+The optimal control problem is given by
 <p>
 <math>
 \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.}
-  & \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}
 </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 ==