Lotka Volterra multi-arcs problem

Lotka Volterra multi-arcs problem
State dimension:	1
Differential states:	3
Discrete control functions:	1
Interior point equalities:	3

This site describes a Lotka Volterra variant with three singular arcs instead of only one in the standard variant.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{ll}\underset{x,w}{\min }& x_2(t_f)\\ \text{s.t.}& {\dot{x}}_0& =& x_0-x_0{x}_1-c_0{x}_{0}w,\\ & {\dot{x}}_1& =& -x_1+x_0{x}_1-c_1{x}_{1}w,\\ & {\dot{x}}_2& =& (x_0-p_0(t){)}^2+(x_1-p_1(t){)}^2,\\ & x(0)& =& (0.5,0.7,0{)}^T,\\ & w(t)& \in & \{0,1\},\end{array}$

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.

$\begin{array}{rcl}[t_0,t_f]& =& [0,18],\\ (c_0,c_1)& =& (0.4,0.2),\\ p_0(t)& =& \{\begin{array}{ll}1& \text{for }t\in [0,7.2],\\ 0.9& \text{for }t\in [7.2,12],\\ 1& \text{for }t\in [12,18].\end{array}\\ p_1(t)& \equiv & p_0(t).\end{array}$

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 $n_t=18000,n_u=300$ is $x_2(t_f)=1.41842699$. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $x_2(t_f)=1.4237735$.

• Reference solution plots
• 
  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=18000,n_u=300$.
• 
  Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=18000,n_u=300$. The relaxed controls were approximated by Combinatorial Integral Approximation.