= Solution
Completing the square in the <Hamilton-Jacobi-Bellman equation> shows that the minimizing feedback is
$$
u^*(x,t)=-\frac{F_x(x,t)}{2c_t}.
$$
This is a <linear-quadratic optimal control> problem, so set $F(x,t)=a(t)x^2$. The equation and terminal condition become
$$
a'(t)=\frac{a(t)^2}{c_t},
\qquad a(h)=1.
$$
This <Riccati equation> is separable and gives
$$
\frac1{a(t)}=1+\int_t^h\frac{ds}{c_s}.
$$
Consequently the general optimal feedback law is
$$
\boxed{u^*(x,t)=-\frac{x}{c_t\left(1+\displaystyle\int_t^h ds/c_s\right)}}.
$$
When $c_t=c$ is constant,
$$
F(x,t)=\frac{c}{c+h-t}x^2,
\qquad
\boxed{u^*(x,t)=-\frac{x}{c+h-t}}.
$$
Along the corresponding optimal trajectory, $x_s=x(c+h-s)/(c+h-t)$, so the open-loop control is constant: $u_s^*=-x/(c+h-t)$ for $s\in[t,h]$.
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