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

17 ビュー (過去 30 日間)
Ehsan Ranjbari
Ehsan Ranjbari 2022 年 1 月 13 日
コメント済み: Ehsan Ranjbari 2022 年 1 月 21 日
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 件のコメント
Torsten
Torsten 2022 年 1 月 13 日
Sorry, but I have no experience with numerical optimal control.
So I can't give you advise in this respect.

サインインしてコメントする。

採用された回答

Torsten
Torsten 2022 年 1 月 14 日
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 件のコメント
Ehsan Ranjbari
Ehsan Ranjbari 2022 年 1 月 21 日
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

サインインしてコメントする。

その他の回答 (1 件)

Walter Roberson
Walter Roberson 2022 年 1 月 13 日
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 件のコメント
Ehsan Ranjbari
Ehsan Ranjbari 2022 年 1 月 14 日
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.

サインインしてコメントする。

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by