The strength of the economy evolves according to the equation

$$\ddot{x}_{t}=-\alpha^{2} x_{t}+u_{t},$$

where $x_{0}=\dot{x}_{0}=0$ and $u_{t}$ is the effort that the government puts into reform at time $t, t \geq 0$. The government wishes to maximize its chance of re-election at a given future time $T$, where this chance is some monotone increasing function of

$$x_{T}-\frac{1}{2} \int_{0}^{T} u_{t}^{2} d t$$

Use Pontryagin's maximum principle to determine the government's optimal reform policy, and show that the optimal trajectory of $x_{t}$ is

$$x_{t}=\frac{t}{2} \alpha^{-2} \cos (\alpha(T-t))-\frac{1}{2} \alpha^{-3} \cos (\alpha T) \sin (\alpha t) .$$