Brachistochrone Problem

The brachistochrone optimal control problem is stated as follows. Minimize the cost functional \[t_f\] subject to the dynamic constraints \[\begin{array}{lcl}\dot{x}(t) & = & v(t)\sin u(t), \\ \dot{y}(t) & = & v(t)\cos u(t), \\ \dot{v}(t) & = & g\cos u(t) \end{array}\] and the boundary conditions \[\begin{array}{lclclcl}x(0) & = & 0 & , & x(t_f) & = & 2, \\ y(0) & = & 0 & , & y(t_f) & = & 2, \\ v(0) & = & 0 & , & x(t_f) & = & \textrm{Free}\end{array}\] The solution to the brachistochrone optimal control problem using GPOPS-II is shown in the figures below.