Exact observability of infinite dimensional port-Hamiltonian systems via the Hautus test

Observability, and in particular exact observability, is an indispensable tool for observer based controller design for infinite dimensional systems. However, determining exact observability for infinite dimensional systems is often very complex. Finite dimensional systems, on the other hand, know many different tools that easily achieve determination of observability. As such, in this work the possibility is explored to extend the utilisation of tools such as the Hautus test, the Crank-Nicolson scheme and the Lyapunov equation to establish exact controllability for a class of infinite dimensional port-Hamiltonian systems. In particular, the goal of this work is to extend the use of the infinite dimensional Hautus test as a sufficient condition for exact observability. This is done by relating the considered continuous-time system to its discrete-time counterpart: this allows, namely, to link the exact observability of the continuous-time system on the one hand to the observability of the discrete-time system on the other. This work finds that, for the considered class of port-Hamiltonian systems (PHS), exact observability may be determined if the discrete-time counterpart of the considered PHS is observable.