Lotka Volterra fishing problem (JuMP)

This is an implementation of the Lotka Volterra fishing problem using JuMP. The problem was discretized and the ODEs are solved using the explicit Euler method. Although not necessary in JuMP the code was divided into three parts (following AMPL) - model file, data file and run file. The run file calls the other files and performs additional tasks such as printing results.

The control function $u(t)$ was relaxed to the domain $[0,1]$.

Model file ("lotka_mod.jl"):

#JuMP implementation of Lotka-Volterra fishing example using collocation
#mod file
 
#declaring the model
m = Model()
 
#defining variables
@defVar(m, x[ii=1:n_x, tt=1:N])
@defVar(m, L_control <= u[tt=1:N] <= U_control)
 
#set objective function
@setObjective(m, Min, x[3,N])
 
#setting constraints
#starting values
@addConstraint(m, starting_value[ii=1:n_x], x[ii,1] == x_start[ii])
 
#ODE - solved with explicit euler method (i.e. x_k+1 = x_k + stepsize * f(x_k, t_k))
@addConstraint(m, ODE[ii=1:n_x, tt=1:N-1],  x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(x[:,tt], u[tt])[ii]  == 0)

Data file ("lotka_dat.jl"):

#JuMP implementation of Lotka-Volterra fishing example using collocation
#dat file
 
#general parameters
c0 = 0.4;
c1 = 0.2;
 
#number of states
n_x = 3;
 
##discretization
#number of shooting intervals / discretization points
N = 300;
#starting / end time
t_start = 0;
t_end = 12;
#time discretization
time = linspace(t_start,t_end, N+1);
step_size = (t_end - t_start)/N;
 
#starting value
x_start = [0.5, 0.7, 0];
 
#bounds for control
L_control = 0
U_control = 1;
 
 
##right hand side of ODE
function ode_rhs(state, control)
#give in form f1, f2, f3,...
 state[1] - state[1] * state[2] - c0 * state[1] * control,
-state[2] + state[1] * state[2] - c1 * state[2] * control,
(state[1]-1) * (state[1]-1) + (state[2]-1) * (state[2]-1)
end

Run file ("lotka_run.jl"):

#JuMP implementation of Lotka-Volterra fishing example using collocation
#run file
 
using JuMP;
using Ipopt;
 
 
println("----------------------------------------------------")
println("Time used for data")
@time include("lotka_dat.jl")
println("----------------------------------------------------")
println("Time used for modeling")
@time include("lotka_mod.jl")
println("----------------------------------------------------")
println("Time used for solving")
@time solve(m);
 
println("----------------------------------------------------")
println("----------------------------------------------------")
 
 
println("Optimal objective value is: ", getObjectiveValue(m))
println("Optimal Solution is: \n", getValue(x), getValue(u))