%   
% Written by:
% -- 
% John L. Weatherwax                2006-08-28
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all; clc; clear; 

% set the constants of this problem: 
u0      = 1.;
x0      = 0.5; % the constant x sub 0
n       = 2;
a       = 0.015;
b       = 0.01; 
c       = 0;

% the times and initial conditions: 
t_0     = 0.; 
t_f     = 10.; 
x_0_ic  = 0.4; % the initial conditions on x(t)

% find the optimal value for the initial condition for lambda(t) or lambda_0 so that lambda(t_f) matches the known boundary:
%  
lambda_0 = fminsearch( @(x) sect_4_prob_2_J_min_fn(x, t_0,t_f,x_0_ic, u0,x0,n,a,b,c), [ -10. ] )

% evaluate J at this value of lambda_0:
% 
sect_4_prob_2_J_min_fn(lambda_0, t_0,t_f,x_0_ic, u0,x0,n,a,b,c)

% plot the state x(t) and the control u(t): 
% 
sol = ode23s( @(t,x) sect_4_prob_2_J_ode_fn(t,x, u0,x0,n,a,b,c), [t_0,t_f], [lambda_0;x_0_ic] );

t      = sol.x;
lambda = sol.y(1,:);
x      = sol.y(2,:);

figure; plot( t, x, '-o' ); 
xlabel('time'); ylabel('x(t)'); 
%saveas( gcf, '../../WriteUp/Graphics/Chapter3/sect_4_prob_2_x_t.eps', 'epsc' );

figure; plot( t, lambda, '-o' ); 
xlabel('time'); ylabel('lambda(t)'); 

u = sect_4_prob_2_u_from_x_N_lambda(t,x,lambda, u0,x0,n,a,b,c);
figure; plot( t, u, '-o' );
xlabel('time'); ylabel('u(t)'); 
%saveas( gcf, '../../WriteUp/Graphics/Chapter3/sect_4_prob_2_u_t.eps', 'epsc' );