State Pontryagin's maximum principle in the case where both the terminal time and the terminal state are given.

Show that $\pi$ is the minimum value taken by the integral

$$\frac{1}{2} \int_{0}^{1}\left(u_{t}^{2}+v_{t}^{2}\right) d t$$

subject to the constraints $x_{0}=y_{0}=z_{0}=x_{1}=y_{1}=0$ and $z_{1}=1$, where

$$\dot{x}_{t}=u_{t}, \quad \dot{y}_{t}=v_{t}, \quad \dot{z}_{t}=u_{t} y_{t}-v_{t} x_{t}, \quad 0 \leqslant t \leqslant 1$$

[You may find it useful to note the fact that the problem is rotationally symmetric about the $z$-axis, so that the angle made by the initial velocity $\left(\dot{x}_{0}, \dot{y}_{0}\right)$ with the positive $x$-axis may be chosen arbitrarily.]