Good afernoon community, recently someone recommends me GEKKO library to do dynamic optimization through python. I'm trying to replicate this paper with that. Basically the model is:
According to the paper E is control variable and K is a state variable, the rest are constant values. I think my error is when I define E as a control variable, because when I do the output is that GEKKO can't find a solution. That's is the reason that belows I define as a parameter.
# Initialize GEKKO
m = GEKKO()
# Define values:
# Constant values
beta_1, beta_2, gamma, delta, s, alpha, L_bar = 0.2, 0.8, 0.5, 0.05, 0.4719, 10, 900000000
# Time period
n = 100
t = np.linspace(0, 15, n)
m.time = t
# Set initial conditions:
# Define K(0) = K_0 initial capital
K = m.Var(value = 0)
# Restrictions problem:
# Define constraints
import math
E = m.Param(-4 * (np.exp(t/2) - np.exp(-1))) # Control variable E
Y = m.Param((1 - gamma) * (K**beta_1) * (L_bar**(beta_2)) + gamma * E)
K_dot = m.SV(1, lb = 0, ub = 1)
m.Equation(K_dot.dt() == s * (Y - E) - delta * K)
J = m.Var(value = 0)
Jf = m.FV()
Jf.STATUS = 1
# Define objective function :
m.Connection(Jf,J,pos2='end')
m.Equation(J.dt() == -1 *(E**2) - alpha * K_dot)
# Set solving options
m.Obj(-Jf) # maximize profit
# Set solving options
m.options.IMODE = 6 # optimal control
m.options.NODES = 3 # collocation nodes
m.options.SOLVER = 3 # solver (IPOPT)
# Solve maximization problem
m.solve(disp = False) # Solve
plt.plot(t, K, "--r")
Taking a look to the documentation, I think my problem is a mix between the "load following" exmple and the "optimal control with economic objective". In order to know if I did in the right way, I use figure 5 and 6 as reference. For the figure 5 when x = 11 there is change in the graphic...but for me it isn't. While in figure 6, the y axis is from [0, -3k] and according to my script is from [0, -7k].