Solution
= Solution
At time $n$, the minimizing feedback from part (b) is
$$
u_n^*=-\frac{X_{n-1}}{N-n+2}.
$$
Starting with $u_1^*=-X_0/(N+1)$ and repeatedly substituting
$X_j=X_{j-1}+u_j^*+\xi_j$ gives by induction
$$
X_{n-1}=\frac{N-n+2}{N+1}X_0
+\sum_{j=1}^{n-1}\frac{N-n+2}{N-j+1}\xi_j.
$$
Therefore
$$
\boxed{u_n^*=-\frac{X_0}{N+1}-\frac{\xi_1}{N}
-\frac{\xi_2}{N-1}-\cdots-\frac{\xi_{n-1}}{N-n+2}},
$$
as required.