%% formulation
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\begin{align*}
\min_{u(t),t_f} \quad & -x(1) \\
\text{subject to:} \quad & \begin{bmatrix}
x^\prime \\ y^\prime \\ v_x^\prime \\ v_y^\prime
\end{bmatrix} = \begin{bmatrix}
t_f v_x \\
t_f v_y \\
\frac{t_f}{m}\left( -L \sin(\eta) - D \cos(\eta) \right) \\
\frac{t_f}{m}\left( L \cos(\eta) - D \sin(\eta) - mg \right)
\end{bmatrix} \\
& (x,y,v_x,v_y)(0) = (0,1000,13.2275675,-1.28750052) \\
& (y,v_x,v_y)(1) = (1000,13.2275675,-1.28750052) \\
& 0 \leq u(t) \leq 1.4 \\
\text{where:} \quad & t_f = p_1 \\
& C_D = C_0 + k u^2 \\
& D = \frac{1}{2} C_D \rho S v_r^2 \\
& L = \frac{1}{2} C_L \rho S v_r^2 \\
& X = \left(\frac{x}{R} - 2.5 \right)^2 \\
& V_y = v_y - u_M\left(1 - X \right) e^{-X} \\
& v_r = \sqrt{v_x^2 + V_y^2} \\
& \sin\eta = \frac{V_y}{v_r} \\
& \cos\eta = \frac{V_x}{v_r}
\end{align*}
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%% solution