LV Shared Resource

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

Mathematical formulation

The optimal control problem is given by

$\begin{array}{llcl}\underset{u}{\min }& \underset{0}{\overset{t_f}{\int }}& & (x_0(t)-1.5{)}^2+(x_1(t)-1{)}^2+(x_2(t)-1{)}^{2}dt\\ \text{s.t.}& {\dot{x}}_0& =& x_0(t)-x_0(t)x_1(t)-x_0(t)x_2(t),\\ & {\dot{x}}_1& =& -x_1(t)+x_0(t)x_1(t)-c_1{x}_1(t)u(t),\\ & {\dot{x}}_2& =& -x_2(t)+\alpha x_0(t)x_2(t)-c_2{x}_2(t)u(t),\\ & x(0)& =& x_0,\\ & u(t)& \in & [0,1],\\ & \alpha & >& 1.\end{array}$

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.