If we have a continuous-time system with a scalar state variable, plant equation $$\dot{x}= u,$$ and cost function $$Q\int_o^h u^2 dt + x(h)^2,$$ then by writing the dynamic programming equation in infinitesimal form and taking a limit, I wish to show that the value function $F$ satisfies
$$ 0 = \frac{\partial{F}}{\partial{t}} + \inf_u \left[Qu^2 + \frac{\partial{F}}{\partial{x}}u\right].$$
I then want to show that $F$ and the optimal control with time $s$ to go are
$$F=\frac{Qx^2}{Q+s}$$ and $$u = - \frac{x}{Q+s}.$$
Any help with this question would be really appreciated. I am finding optimisation in continuous time very difficult.