How to find a proper algorithm to solve this optimal control problem?

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Hi everyone!
I am trying to find a way to solve this optimal control problem in MATLAB. The function is too complex and the using Hamiltonian in MATLAB couldn't help.The problem describes as below:
p = 100;
a = 0;
b = 0.07;
c = 0.04;
r = 0.005;
z = 0.1;
c0 = 70;
x0 = 0.4;
alpha = 0.005;
beta = 0.006;
gamma = 0.003;
delta = 0.007;
Dx = (alpha + beta*u + (gamma + delta*u)*x)*(1-x); % State Equation
f = ((p - c0*((x0/x)^z))*Dx) - (a + (b*u) + (c*u^2)); % Function inside the integral (Cost function)
% x(t0) = 0.4, x(tf) = free, t0 = 0. tf = 31
Note that the aim is to maximize the function f.
I tried to use fmincon and still the function is too complex to get an answer.
Thanks!
  3 Commenti
Ehsan Ranjbari
Ehsan Ranjbari il 13 Gen 2022
Thanks for your kind reply.
I would also want to know if it is a good idea to first linearize the cost function and then using the Hamiltonian?
thanks in advance.
Torsten
Torsten il 13 Gen 2022
Sorry, but I have no experience with numerical optimal control.
So I can't give you advise in this respect.

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Risposta accettata

Torsten
Torsten il 14 Gen 2022
This should give you a start:
%Optimal advertising expenditure in monopoly
%% Constants
p = 100;
a = 0;
b = 0.07;
c = 0.04;
r = 0.005;
z = 0.1;
c0 = 70;
x0 = 0.4;
alpha = 0.005;
beta = 0.006;
gamma = 0.003;
delta = 0.007;
%% State equation (g)
syms x u p1
Dx = (alpha + beta*u + (gamma + delta*u)*x)*(1-x);
%% Cost function inside the integral (f)
f = ((p - c0*((x0/x)^z))*Dx) - (a + (b*u) + (c*u^2));
%% Hamiltonian %lambda_0= 1 (Normal case)
H = f + p1*Dx;
%% Costate equations
Dp1 = -diff(H,x);
%% solve for control u
du = diff(H,u);
sol_u = solve(du,u);
f = subs(f,u,sol_u)
Dp1 = subs(Dp1,u,sol_u)
rhs = [f;Dp1];
% Turn to numerical computation
fun = matlabFunction(rhs)
tmesh = linspace(0,31,150);
guess = @(x)[0.4*(1-x/31)+x/31;1]
solinit = bvpinit(tmesh,guess);
bvpfcn = @(t,y)fun(y(2),y(1));
bcfcn = @(ya,yb)[ya(1)-0.4;yb(1)-1];
sol = bvp4c(bvpfcn, bcfcn, solinit)
  2 Commenti
Torsten
Torsten il 14 Gen 2022
Since you want to maximize, you'll have to take
f = -(((p - c0*((x0/x)^z))*Dx) - (a + (b*u) + (c*u^2)));
I guess.
Ehsan Ranjbari
Ehsan Ranjbari il 21 Gen 2022
I managed to reformulate the problem and actually simplified it. Now the problem reduces to this: https://www.mathworks.com/matlabcentral/answers/1630720-solve-optimal-control-problem-with-free-final-time-using-matlab?s_tid=srchtitle
I would love to hear your suggestion on solving this.
Thanks in Advance

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Più risposte (1)

Walter Roberson
Walter Roberson il 13 Gen 2022
You did not say what you wanted to optimzie with respect to. If you wanted to optimize with respect to u, then see solu below.
If you wanted to optimize with respect to x (in terms of u) then I will need to do more testing.
syms x u
p = 100;
a = 0;
b = 0.07;
c = 0.04;
r = 0.005;
z = 0.1;
c0 = 70;
x0 = 0.4;
alpha = 0.005;
beta = 0.006;
gamma = 0.003;
delta = 0.007;
Dx = (alpha + beta*u + (gamma + delta*u)*x)*(1-x); % State Equation
f = ((p - c0*((x0/x)^z))*Dx) - (a + (b*u) + (c*u^2)); % Function inside the integral (Cost function)
f
f = 
Dfu = diff(f,u)
Dfu = 
string(Dfu)
ans = "((7*x)/1000 + 3/500)*(70*(2/(5*x))^(1/10) - 100)*(x - 1) - (2*u)/25 - 7/100"
solu = simplify(solve(Dfu, u))
solu = 
Dfx = diff(f,x)
Dfx = 
string(Dfx)
ans = "(70*(2/(5*x))^(1/10) - 100)*((3*u)/500 + x*((7*u)/1000 + 3/1000) + 1/200) + ((7*u)/1000 + 3/1000)*(70*(2/(5*x))^(1/10) - 100)*(x - 1) - (14*(x - 1)*((3*u)/500 + x*((7*u)/1000 + 3/1000) + 1/200))/(5*x^2*(2/(5*x))^(9/10))"
  3 Commenti
Torsten
Torsten il 13 Gen 2022
Modificato: Torsten il 13 Gen 2022
I think the problem is
Find u(t) such that
integral_{t=0}^{t=tf} ((p - c0*((x0/x(t))^z))*dx/dt) - (a + (b*u(t)) + (c*u(t)^2)) dt
is maximized under the constraint
dx/dt = (alpha + beta*u(t) + (gamma + delta*u(t))*x(t))*(1-x(t))
x(0)=0, x(tf) = free
Ehsan Ranjbari
Ehsan Ranjbari il 14 Gen 2022
Thank you both for the comments.
Yes. The problem is exactly the on @Torsten mentioned above.
This problem is generally an economic one and the aim is to maximize the that integral (market share).
I want to share my effort on using Hamiltonian:
%Optimal advertising expenditure in monopoly
%% Constants
p = 100;
a = 0;
b = 0.07;
c = 0.04;
r = 0.005;
z = 0.1;
c0 = 70;
x0 = 0.4;
alpha = 0.005;
beta = 0.006;
gamma = 0.003;
delta = 0.007;
%% State equation (g)
syms x u p1;
Dx = (alpha + beta*u + (gamma + delta*u)*x)*(1-x);
%% Cost function inside the integral (f)
syms f ;
f = ((p - c0*((x0/x)^z))*Dx) - (a + (b*u) + (c*u^2));
%% Hamiltonian %lambda_0= 1 (Normal case)
syms H p1 ;
H = f + p1*Dx;
%% Costate equations
Dp1 = -diff(H,x);
%% solve for control u
du = diff(H,u);
sol_u = solve(du,u);
%% Substitute u to state equation
Dx = subs(Dx, u, sol_u);
f = subs(f, u, sol_u);
%% Convert symbolic objects to strings for using 'dsolve'
eq1 = strcat('Dx=',char(Dx));
eq2 = strcat('Dp1=',char(Dp1));
sol_h = dsolve(eq1,eq2);
%% Use boundary conditions to determine the coefficients
% conA1 = 'x(0) = 0.4';
% conA2 = 'x(31) = 1';
% sol_a = dsolve(eq2,conA1,conA2);
after running this I get the warning for eq1 which is:
% Warning: Unable to find symbolic solution.
I think it is better to use the numerical solutions and see what happens.

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