Conservative Control Systems Described by the Schrödinger Equation

An important subclass of well-posed linear systems is formed by the conservative systems. A conservative system is a system for which a certain energy balance equation is satisfied both by its trajectories and those of its dual system. In Malinen et al. [10], a number of algebraic characterizations of conservative linear systems are given in terms of the operators appearing in the state space description of the system. Weiss and Tucsnak [20] identified by a detailed argument a large class of conservative linear systems described by a second order differential equation in a Hilbert space and an output equation, and they may have unbounded control and observation operators. In this paper, we give two examples of conservative linear control systems described by the linear Schrödinger equation on an n-dimensional domain with boundary control and boundary observation. These examples do not fit into the framework of [20].