= Solution
The <dynamic programming> principle over a short interval $[t,t+\Delta t]$ gives
$$
F(x,t)=\inf_u\left\{c_tu^2\Delta t+F(x+u\Delta t,t+\Delta t)\right\}+o(\Delta t).
$$
A first-order <Taylor expansion> of the <value function> gives
$$
F(x+u\Delta t,t+\Delta t)
=F(x,t)+F_xu\Delta t+F_t\Delta t+o(\Delta t).
$$
After subtracting $F(x,t)$, dividing by $\Delta t$ and taking the <limit>, the <Hamilton-Jacobi-Bellman equation> is
$$
\boxed{F_t=-\inf_u\{c_tu^2+F_xu\}},
\qquad F(x,h)=x^2.
$$
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