 Terminal-position reduction of double-integrator control

For $dx=ydt$, $dy=udt+dW$ and terminal cost $x(T{)}^2$ plus running cost $u^2$, the displayed change of state gives $dz=(T-t)udt+(T-t)dW$. A quadratic value function has coefficient $P(t)=[1+(T-t{)}^3/3{]}^{-1}$ and additive noise cost $\log [1+(T-t{)}^3/3]$. Minimizing its Hamilton-Jacobi-Bellman equation gives the displayed control. The gain is unchanged if the additive noise is removed, illustrating certainty equivalence.