function [J] = sect_5_prob_6_J_min_fn(lambda_0, t_0, t_f, x10,x20,T1,T2,xair,xd,k1,k2,c1,c2)
%   
% 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 = ode23s( @(t,x) sect_5_prob_6_ode_fn(t,x, T1,T2,xair,xd,k1,k2,c1,c2), [t_0,t_f], [x10;x20;lambda_0(1);lambda_0(2)] );

x1_t      = sol.y(1,:); % extract x_1(t)
x2_t      = sol.y(2,:); % extract x_2(t)
lambda1_t = sol.y(3,:); % extract lambda_1(t)
lambda2_t = sol.y(4,:); % extract lambda_2(t)

% Second evaluate the expression for J (the final conditions on lambda1_t and lambda2_t should be zero): 
% 
J = 0.5 * ( ( lambda1_t(end) - 2*c1*(x1_t(end)-xd) )^2 + ( lambda2_t(end) - 2*c1*(x2_t(end)-xd) )^2 );