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

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!

[Torsten]

It will cost you quite a lot of effort to program an optimal control problem from scratch.

If you already spent this effort, please show your attempt with "fmincon".

Otherwise you should use one of the control packages which use MATLAB, e.g.

https://tomopt.com/tomlab/products/propt/

[Ehsan Ranjbari]

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]

Sorry, but I have no experience with numerical optimal control.

So I can't give you advise in this respect.

Answer 1

[Torsten]

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)

[Torsten]

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]

Dear @Torsten

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

Answer 2

[Walter Roberson]

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))"

[Walter Roberson]

Solving the derivative with respect to x gives you

1 / RootOf((42*u+35)*Z^21+(-63*u+126)*Z^11+(50*5^(1/10)*2^(9/10)*u-100*2^(9/10)
*5^(1/10))*Z^10+(931*u+399)*Z-700*5^(1/10)*2^(9/10)*u-300*2^(9/10)*5^(1/10), Z)^
10

That is, you have to solve the degree 21 polynomial involving u to find the set of values, Z, such that the expression becomes 0; and when you do, the -10'th power of that is the optimal x.

Polynomials of degree 21 do not generally have algebraic solutions, so you would have to solve numerically for each different u value

[Torsten]

@Walter Roberson

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]

@Walter Roberson and @Torsten.

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.