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Interpret unit-intensity Gaussian white noise as $d{W}_t/dt$. Differentiating $z=x+(T-t)y$ cancels the two $ydt$ terms and givesThe cost is terminal $z(T{)}^2$ plus the same running $u^2$. Write $h=T-t$ and guess the value $V(t,z)=P(t)z^2+C(t)$. The Hamilton-Jacobi-Bellman equation isCompleting the square gives $u_{\ast }=-hPz$. Equating powers yields $P^{\prime }=h^2{P}^2$, $C^{\prime }=-h^{2}P$, with $P(T)=1,C(T)=0$. Integration givesIndeed $V_t+{\mathcal{L}}^{u}V+u^2=(u+hPz{)}^2\ge 0$, and the Itô verification identity attains equality at this control. This proves optimality for admissible controls of finite expected quadratic cost. The additive noise changes only $C$, not the feedback gain, so this is certainty-equivalence control. A different constant noise intensity multiplies $C$ but leaves the control unchanged, as in the terminal-position reduction of double-integrator control.