Consider the deterministic system

$$\dot{x}_{t}=u_{t}$$

where $x_{t}$ and $u_{t}$ are scalars. Here $x_{t}$ is the state variable and the control variable $u_{t}$ is to be chosen to minimise, for a fixed $h>0$, the cost

$$x_{h}^{2}+\int_{0}^{h} c_{t} u_{t}^{2} \mathrm{~d} t$$

where $c_{t}$ is known and $c_{t}>c>0$ for all $t$. Let $F(x, t)$ be the minimal cost from state $x$ and time $t$.

(a) By writing the dynamic programming equation in infinitesimal form and taking the appropriate limit show that $F(x, t)$ satisfies

$$\frac{\partial F}{\partial t}=-\inf _{u}\left[c_{t} u^{2}+\frac{\partial F}{\partial x} u\right], \quad t<h$$

with boundary condition $F(x, h)=x^{2}$.

(b) Determine the form of the optimal control in the special case where $c_{t}$ is constant, and also in general.