Time-varying additive perturbations of well-posed linear systems

We study a time-varying well-posed system resulting from the additive perturbation of the generator of a time-invariant well-posed system. The associated generator family has the form A+G(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A+G(t)$$\end{document}, where G(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(t)$$\end{document} is a bounded operator on the state space and G(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\cdot )$$\end{document} is strongly continuous. We show that the resulting time-varying system (the perturbed system) is well-posed and investigate its properties. In the particular case when the unperturbed system is scattering passive, we derive an energy balance inequality for the perturbed system. We illustrate this theory using it to formulate the system corresponding to an electrically conducting rigid body moving in vacuo in a bounded domain, with an electromagnetic field (both in the rigid body and in the vacuum) described by Maxwell’s equations.