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The minimizer in the quadratic expression from part (b) isRelabelling the time index,

If $\mathrm{Var}(x_0)$ or the observation-noise variances change, the Kalman filter gains and the sequence $V_t$ change. The control gain $K_t$ does 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.