A semigroup characterization of well-posed linear control systems

We study the well-posedness of a linear control system Σ(A,B,C,D) with unbounded control and observation operators. To this end we associate to our system an operator matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} on a product space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{X}^{p}$\end{document} and call it p-well-posed if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} generates a strongly continuous semigroup on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{X}^{p}$\end{document}. Our approach is based on the Laplace transform and Fourier multipliers. The results generalize and complement those of Curtain and Weiss (Int. Ser. Numer. Math. vol. 91. Birkhäuser, Basel, 1989), Staffans and Weiss (Trans. Am. Math. Soc. 354:3229–3262, 2002) and are illustrated by a heat equation with boundary control and point observation.