Stability Analysis of the Observer Error of an In-Domain Actuated Vibrating String

In this paper, the behaviour of the observer error of an in-domain actuated vibrating string, where the observer system has been designed based on energy considerations exploiting a port-Hamiltonian system representation for infinite-dimensional systems, is analysed. Thus, the observer-error dynamics are reformulated as an abstract Cauchy problem, which enables to draw conclusions regarding the well-posedness of the observer-error system. Furthermore, we show that the observer error is asymptotically stable by applying LaSalle's invariance principle for infinite-dimensional systems.