Observer-Based Output-Feedback Stabilization of Continua of Linear Hyperbolic PDEs Using Backstepping

We develop a boundary observer design for a class of continua of linear hyperbolic PDE systems, which are viewed as the continuum version of n + m general heterodirectional hyperbolic systems as n → ∞. The design relies on the introduction of a novel, continuum PDE backstepping transformation, which enables the construction of a Lyapunov functional for the estimation error system. We then introduce the respective non-collocated, output-feedback design employing the stabilizing continuum control kernels from Humaloja and Bekiaris-Liberis [2024]. (The observer-based output-feedback stabilization problem for the class of ∞ + 1 hyperbolic systems can be solved as special case of the design procedure presented here.) Stability under the observer-based output-feedback law is established by using the Lyapunov functional construction for the estimation error system and proving well-posedness of the complete closed-loop system, which allows utilization of the separation principle. We illustrate the output-feedback design in simulation, via a numerical example for which the control and observer kernels can be computed in closed form.