Conservative boundary control systems

We study continuous time linear dynamical systems of boundary control/observation type, satisfying a Green–Lagrange identity. Particular attention is paid to systems which have a well-defined dynamics both in the forward and the backward time directions. As we change the direction of time we also interchange inputs and outputs. We show that such a boundary control/observation system gives rise to a continuous time Livšic–Brodski (system) node with strictly unbounded control and observation operators. The converse is also true. We illustrate the theory by a classical example, namely, the wave equation describing the reflecting mirror.