= Terminal-position reduction of double-integrator control
{title2=$z=x+(T-t)y,\qquad u_*=-\frac{(T-t)z}{1+(T-t)^3/3}$}
For $dx=y\,dt$, $dy=u\,dt+dW$ and terminal cost $x(T)^2$ plus running cost $u^2$, the displayed change of state gives $dz=(T-t)u\,dt+(T-t)dW$. 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.
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