Lotka Volterra absolute fishing problem

Lotka Volterra absolute fishing problem
State dimension:	1
Differential states:	3
Discrete control functions:	5
Interior point equalities:	3

This site describes a Lotka Volterra variant with five binary controls that all represent fishing of an absolute biomass.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{llclr} \min_{x, w} & x_{2} (t_{f}) \\ s.t. & {\dot{x}}_{0} & = & x_{0} - x_{0} x_{1} - \sum_{i = 1}^{5} c_{0, i} w_{i}, \\ {\dot{x}}_{1} & = & - x_{1} + x_{0} x_{1} - \sum_{i = 1}^{5} c_{1, i} w_{i}, \\ {\dot{x}}_{2} & = & (x_{0} - 1)^{2} + (x_{1} - 1)^{2}, \\ x (0) & = & (0.5, 0.7, 0)^{T}, \\ \sum_{i = 1}^{5} w_{i} (t) & = & 1, \\ w_{i} (t) & \in & {0, 1}, i = 1 \dots 5 . \end{array}$

Here the differential states $(x_{0}, x_{1})$ 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 $\min x_{2} (t_{f})$ . This problem variant allows to choose between five different fishing options.

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.