fstar( x, u, option )

function [ sys ] = fstar( x, u, option )
% This function must be created  by the user to define
% the dynamic system xdot = f*(x,u,t)


if  option == 'value' 

% return the value xdot from
% xdot = f*(x,u,t)

   sys = zeros(2,1);

   sys(1) = -(x(1)+0.25)+(x(2)+0.5)*exp(25*x(1)/(x(1)+2))-(1+u(1))*(x(1)+0.25);
   sys(2) = 0.5-x(2)-(x(2)+0.5)*exp(25*x(1)/(x(1)+2));


elseif  option == 'gradz' 

% return the jacobian matrix of f*(x,u,k) with respect to x 

     sys = zeros( 2,2 );

     h = [sqrt(eps); 0];

     sys(:,1) = (fstar(x+h,u,'value')-fstar(x-h,u,'value'))/(2*norm(h));

     h = [0; sqrt(eps)];

     sys(:,2) = (fstar(x+h,u,'value')-fstar(x-h,u,'value'))/(2*norm(h));

elseif  option == 'gradv' 

% return the jacobian matrix of f*(x,u,k) with respect to u

          h = sqrt(eps);

          sys = (fstar(x,u+h,'value')-fstar(x,u-h,'value'))/(2*h);         

elseif  option == 'xinit' 

% return the initial state conditons

          sys = [ 0.05; 0 ];
%          sys = [ 0.09; 0.09 ];


else sys = [];

end;

return;