On the infinitesimal generator of an optimal state semigroup

TL;DR

This paper precisely characterizes the domain of the infinitesimal generator of the optimal state semigroup in boundary control systems, improving previous partial results by establishing an exact equality through advanced asymptotic and regularity analysis.

Contribution

It provides a complete description of the generator's domain for a class of boundary control systems, advancing the theoretical understanding of the associated optimal control semigroup.

Findings

Exact domain characterization of the generator achieved

Asymptotic estimates crucial for the proof

Validation of parameter relations in coupled PDE systems

Abstract

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.