  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 key
part of the proof of this result developes through appropriate asymptotic
estimates that take advantage of the regularity analysis previously performed
in the study of the optimization problem, while the powers of positive
operators, interpolation and semigroups provide the analytical 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.
