  Late-lumping feedback design for infinite-dimensional linear systems with
unbounded input operators is considered. The proposed scheme is suitable for
the approximation of backstepping and flatness-based designs and relies on a
decomposition of the feedback into a bounded and an unbounded part.
Approximation applies to the bounded part only, while the unbounded part is
assumed to allow for an exact realization. Based on spectral results, the
convergence of the closed-loop dynamics to the desired dynamics is established.
By duality, similar results apply to the approximation of the observer
output-injection gains for systems with boundary observation. The proposed
design and approximation steps are demonstrated and illustrated based on a
hyperbolic infinite-dimensional system.
