Consider the problem

$$\operatorname{minimize} E\left[x(T)^{2}+\int_{0}^{T} u(t)^{2} d t\right]$$

where for $0 \leqslant t \leqslant T$,

$$\dot{x}(t)=y(t) \text { and } \quad \dot{y}(t)=u(t)+\epsilon(t),$$

$u(t)$ is the control variable, and $\epsilon(t)$ is Gaussian white noise. Show that the problem can be rewritten as one of controlling the scalar variable $z(t)$, where

$$z(t)=x(t)+(T-t) y(t) .$$

By guessing the form of the optimal value function and ensuring it satisfies an appropriate optimality equation, show that the optimal control is

$$u(t)=-\frac{(T-t) z(t)}{1+\frac{1}{3}(T-t)^{3}} .$$

Is this certainty equivalence control?