function [J] = sect_3_prob_4_J_min_fn(alpha, T, t_0, t_f, ... 
                                      v_0, gamma_0, h_0, ... % initial conditions
                                      v_d, gamma_d, h_d, ... % desired end point conditions 
                                      q_v, q_gamma, q_h, r_1, r_2) % 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_4_J_ode_fn(t,x, alpha,T), [t_0,t_f], [v_0;gamma_0;h_0] );

v_t = sol.y(1,:);
gamma_t = sol.y(2,:); % extract gamma(t) 
h_t = sol.y(3,:); 

% Second evaluate the expression for J: 
% 
J_part1 = q_v * ( v_t(end) - v_d )^2;
J_part2 = q_gamma * ( gamma_t(end) - gamma_d )^2;
J_part3 = q_h * ( h_t(end) - h_d )^2;
J_part4 = r_1 * alpha^2 * t_f + r_2 * T^2 * t_f;
J       = 0.5 * ( J_part1 + J_part2 + J_part3 + J_part4 );