A state variable $x=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$ is subject to dynamics

$$\begin{aligned} &\dot{x}_{1}(t)=x_{2}(t) \\ &\dot{x}_{2}(t)=u(t) \end{aligned}$$

where $u=u(t)$ is a scalar control variable constrained to the interval $[-1,1]$. Given an initial value $x(0)=\left(x_{1}, x_{2}\right)$, let $F\left(x_{1}, x_{2}\right)$ denote the minimal time required to bring the state to $(0,0)$. Prove that

$$\max _{u \in[-1,1]}\left\{-x_{2} \frac{\partial F}{\partial x_{1}}-u \frac{\partial F}{\partial x_{2}}-1\right\}=0$$

Explain how this equation figures in Pontryagin's maximum principle.

Use Pontryagin's maximum principle to show that, on an optimal trajectory, $u(t)$ only takes the values 1 and $-1$, and that it makes at most one switch between them.

Show that $u(t)=1,0 \leqslant t \leqslant 2$ is optimal when $x(0)=(2,-2)$.

Find the optimal control when $x(0)=(7,-2)$.