Lotka Volterra Multimode fishing problem

From mintOC
Jump to: navigation, search
Lotka Volterra Multimode fishing problem
State dimension: 1
Differential states: 3
Discrete control functions: 3
Interior point equalities: 3

This site describes a Lotka Volterra variant with three binary controls instead of only one control.

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.} 
 & \dot{x}_0 & = &  x_0 - x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{0,i}\; x_0 \; w_i, \\
 & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; \sum\limits_{i=1}^{3} c_{1,i}\; x_1 \; 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}^{3}w_i(t) &=& 1, \\
 & w_i(t) &\in&  \{0, 1\}, \quad i=1\ldots 3.
\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 three 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).
\end{array}

Reference Solutions

If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, 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=400  is x_2(t_f) =1.82875272. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is x_2(t_f) =1.82878681.