Optimal control problem with integral cost function and both initial and terminal conditions

Hello everybody, I would like to implement in Matlab an optimal control theory problem I am studying for my thesis. The problem is quite complex, I will attach it here:

Minimize the cost functional J on the interval [0,T]

subject to

with constraints and boundary data

I didn't know where to start solving this problem, because of the mixed initial and terminal conditions, I searched online and I started from simple linear problems, managing to resolve them. But when I try to complicate things, inserting nonlinear dynamics I get an error, in particular it says 'Dot indexing is not supported for variables of this type'. I attach the code to explain how I am solving this simple problems, but my question is if the error is due to something I can fix or I have to change my strategy. In this case it would be really helpful to know to deal with it.

The code I am attaching (and for which I get the error) is for solving the problem:

with J(u) integral cost function and initial and terminal constraints

% State equations
syms x1 x2 p1 p2 u;

%dynamics 
%Dx1=x2;
Dx1 = x1*x2; %non lineare 
Dx2 = -x2 + u;

% Cost function inside the integral
syms g;
g = 0.5*u^2;
% Hamiltonian
syms p1 p2 H;
H = g + p1*Dx1 + p2*Dx2;
% Costate equations
Dp1 = -diff(H,x1); %creo il sistema 
Dp2 = -diff(H,x2);
% solve for control u
du = diff(H,u);
sol_u= solve(du,u);

% Substitute u to state equations
Dx2 = subs(Dx2, u, sol_u);
% convert symbolic objects to strings for using ’dsolve’
eq1 = strcat('Dx1=',char(Dx1)); %concatena le stringhe 
eq2 = strcat('Dx2=',char(Dx2));
eq3 = strcat('Dp1=',char(Dp1));
eq4 = strcat('Dp2=',char(Dp2));
sol_h = dsolve(eq1,eq2,eq3,eq4);

conA1 = 'x1(0) = 0';
conA2 = 'x2(0) = 0';
conA3 = 'x1(2) = 5';
conA4 = 'x2(2) = 2';
sol_a = dsolve(eq1,eq2,eq3,eq4,conA1,conA2,conA3,conA4);

figure
ezplot(sol_a.x1,[0 2]); %x1 
hold on 
ezplot(sol_a.x2,[0,2]); %x2
ezplot(-sol_a.p2,[0,2]); %controllo 
axis([0 2 -1 7])
legend('x1','x2','u')

Thanks to anyone who will pay some attention to my problem.

Answer 1

[Torsten]

I think this part

% convert symbolic objects to strings for using ’dsolve’
eq1 = strcat('Dx1=',char(Dx1)); %concatena le stringhe 
eq2 = strcat('Dx2=',char(Dx2));
eq3 = strcat('Dp1=',char(Dp1));
eq4 = strcat('Dp2=',char(Dp2));
sol_h = dsolve(eq1,eq2,eq3,eq4);
conA1 = 'x1(0) = 0';
conA2 = 'x2(0) = 0';
conA3 = 'x1(2) = 5';
conA4 = 'x2(2) = 2';
sol_a = dsolve(eq1,eq2,eq3,eq4,conA1,conA2,conA3,conA4);

is equivalent to

syms x1(t) x2(t) p1(t) p2(t)
eq1 = diff(x1,t) == x1*x2;
eq2 = diff(x2,t) == -p2-x2;
eq3 = diff(p1,t) == -p1*x2;
eq4 = diff(p2,t) == -p1*x1 + p2;
eq = [eq1,eq2,eq3,eq4];
conA1 = x1(0)==0;
conA2 = x2(0)==0;
conA3 = x1(2)==5;
conA4 = x2(2)==2;
con = [conA1,conA2,conA3,conA4];
sol_a = dsolve(eq,con)

Is MATLAB able to find a solution ?

I guess no because of the nonlinear terms.

But maybe you can check out "bvp4c" for a numerical solution.

[Giulia Pucci]

Thank you, I will try with bvp4c