function [J] = sect_3_prob_5_J_min_fn(c_1, c_2, t_0, t_f, ... 
                                      x_1d, ... % desired end point conditions 
                                      q, r, omega_n, xi) % scale factors to weight different parts of J
%   
% 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.
% 
%-----

% First integate the given differential equations from t=0 to t=t_f: 
% 
sol = ode45( @(t,x) sect_3_prob_5_J_ode_fn(t,x, c_1,c_2, omega_n,xi,x_1d), [t_0,t_f], [0;0] );

t_t  = sol.x;
x_1t = sol.y(1,:); % extract x_1(t)
x_2t = sol.y(2,:); 
u_t = -c_1 * ( x_1t - x_1d ) - c_2 * x_2t; % compute u(t) 

% Second evaluate the expression for J: 
% 
J_part1 = q * ( x_1t(end) - x_1d )^2;
J_part2 = r * trapz( t_t, u_t.^2 );
J       = 0.5 * ( J_part1 + J_part2 );