Corrections and extensions of "Optimal control of linear systems with almost periodic input" by G. Da Prato and A. Ichikawa

G. Da Prato and A. Ichikawa consider in their 1987 paper the optimal control problem for linear systems with almost periodic inputs. In order to get their results they use the space $B_{ap}^2(H)$, the completion of the space of almost periodic functions in the quadratic mean. They used that $B_{ap}^2(U)$-inputs give almost periodic outputs. By means of a scalar example we show that this does not hold in general. We prove that one of their main results still holds. Furthermore, we extend this result to all forcing terms in $B_{ap}^2(H)$ and show that if the forcing term is almost periodic, then the optimal control is almost periodic as well