  In this article we fully describe the domain of the infinitesimal generator
of the optimal state semigroup which arises in the theory of the
linear-quadratic problem for a specific class of boundary control systems. This
represents an improvement over earlier work of the authors, joint with
Lasiecka, where a set inclusion was established, but not an equality. The novel
part of the proof of this result developes through appropriate asymptotic
estimates that take advantage of the regularity analysis carried out in the
study of the optimization problem, while the powers of positive operators and
interpolation are still key tools. We also attest to the validity of an assumed
relation between two significant parameters in the case of distinct systems of
coupled hyperbolic-parabolic partial differential equations which are pertinent
to the underlying framework.
