How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance

Let A₀ be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C₀ be a bounded operator from D(A^1/2₀) to another Hilbert space U. We prove that the system of equations (Formula Presented) determines a well-posed linear system with input u and output y. The state of this system is (Formula Presented) where X is the state space. Moreover, we have the energy identity (Formula Presented) We show that the system described above is isomorphic to its dual, so that a similar energy identity holds also for the dual system and hence, the system is conservative. We derive various other properties of such systems and we give a relevant example: a wave equation on a bounded n-dimensional domain with boundary control and boundary observation on part of the boundary.