changing the value function

Question

Please see the following code. How can I modify J(i,1) if value functional involves only u instead of u^2? (simply u gives error as the dimension does not match.

function eg2OCmymodel_Descent

eps = 1e-2;

options = odeset('RelTol', 1e-4, 'AbsTol',[1e-4]);

t0 = 0; 

tf = 60; 

step = .4; 

t_segment = 100; 

Tu = linspace(t0, tf, t_segment);    

u = ones(1,t_segment);               

initx = [100];                    

initp = [0];                       

max_iteration = 100;                 

for i = 1:max_iteration
  
   [Tx,X] = ode45(@(t,x) stateEq(t,x,u,Tu), [t0 tf], initx, options); 
   x1 = X(:,1); 
   [Tp,P] = ode45(@(t,p) costateEq(t,p,u,Tu,x1,Tx), [tf t0], initp, options); 
   p1 = P(:,1); 
   p1 = interp1(Tp,p1,Tx);  
   dH = pH(x1,p1,Tx,u,Tu); 
   H_Norm = dH'*dH; 
 
   J(i,1) = 60*((6*(x1')*x1)/length(Tx)-.5*u*(u')/length(Tu)); 
   
   

   if H_Norm < eps
       
       J(i,1)
       break;
   else
       % adjust control for next iteration
       u_old = u;
       u = AdjControl(dH,Tx,u_old,Tu,step); %this function is defined later
   end; 
end

% plot the state variables & cost for each iteration

figure(1);

plot(Tx, X ,'-');

hold on;

plot(Tu,u,'r:');

text(.2,0.08,'x_1(t)');

text(.2,.4, 'u(t)');

s = strcat('Final cost is: J=',num2str(J(end,1)));

text(.4,1,s);

xlabel('time');

ylabel('states');

hold off;

figure(2);

plot(J,'x-');

xlabel('Iteration number');

ylabel('J');

if i == max_iteration

    disp('Stopped before required residual is obtained.');

end



function dx = stateEq(t,x,u,Tu)

dx = zeros(1,1); 

u = interp1(Tu,u,t); 

dx(1) = -u; 


% Costate equations

function dp = costateEq(t,p,u,Tu,x1,xt)

dp = zeros(1,1);

x1 = interp1(xt,x1,t);   % Interploate the state varialbes

u = interp1(Tu,u,t);     % Interploate the control

dp(1) = 0;


% Partial derivative of H with respect to u

function dH = pH(x1,p1,tx,u,Tu)

% interploate the control

u = interp1(Tu,u,tx);

dH = 6-u-p1;


% Adjust the control

function u_new = AdjControl(pH,tx,u,tu,step)

% interploate dH/du

pH = interp1(tx,pH,tu);

u_new = u - step*pH;

changing the value function

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