Impedance passive and conservative boundary control systems (Q998127)

The external Cayley transform is used for the conversion between the linear dynamical systems in scattering form and in impedance form. This transformation is used to define a class of formal impedance conservative boundary control systems (colligations), without assuming a priori that the associated Cauchy problems are solvable. Sufficient and necessary conditions are developed guaranteeing that impedance conservative colligations are internally well-posed boundary nodes, that is, the associated Cauchy problems are solvable and governed by \(C_0\)-semigroups. Further, a ``strong'' variant of colligations is defined and it is shown that ``strong'' impedance conservative boundary colligations are a slight generalization of the ``abstract boundary space'' construction for symmetric operators. Examples, involving the transmission line and the wave equations, are illustrating the obtained results.