Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 30K / c / Solution

At time $n$, the minimizing feedback from part (b) isStarting with $u_1^{\ast }=-X_0/(N+1)$ and repeatedly substituting
$X_j=X_{j-1}+u_j^{\ast }+{\xi }_j$ gives by inductionThereforeas required.