Observer-Based Output-Feedback Backstepping Stabilization of Continua of Hyperbolic PDEs and Application to Large-Scale $n+m$ Coupled Hyperbolic PDEs

We develop a non-collocated, observer-based output-feedback law 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 $n\to\infty$. 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. 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. Motivated by the fact that the continuum-based designs may provide computationally tractable control laws for large-scale, $n+m$ systems, we then utilize the control/observer kernels and the observer constructed for the continuum system to introduce an output-feedback control design for the original $n+m$ system. We establish exponential stability of the resulting closed-loop system, which consists of a mixed $n+m$-continuum PDE system (comprising the plant-observer dynamics), introducing a virtual continuum system with resets, which enables utilization of the continuum approximation property of the solutions of the $n+m$ system by its continuum counterpart (for large $n$). We illustrate the potential computational complexity/flexibility benefits of our approach via a numerical example of stabilization of a large-scale $n+m$ system, for which we employ the continuum observer-based controller, while the continuum-based stabilizing control/observer kernels can be computed in closed form.