= Solution
The minimizer in the quadratic expression from part (b) is
$$
u_{t-1}^*
=-\frac{P_t}{1+P_t}\widehat x_{t-1}.
$$
Relabelling the time index,
$$
\boxed{u_t^*=-K_t\widehat x_t,\qquad
K_t=\frac{P_{t+1}}{1+P_{t+1}}
=\frac1{T-t+1}.}
$$
If $\operatorname{Var}(x_0)$ or the observation-noise variances change, the <Kalman filter> gains and the sequence $V_t$ change. The control gain $K_t$ does \b[not] change, because the state dynamics and quadratic cost are unchanged. This is the <separation principle>: estimation uncertainty changes the additive constants $d_t$, while the optimal linear feedback gain is the full-information gain.
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