Stabilizing Output-Feedback Control Law for Hyperbolic Systems Using a Fredholm Transformation

For most networked systems found in the literature, the actuated boundary is usually located at one end. In this article, we first consider the stabilization of a chain of two interconnected subsystems, actuated at the in-between boundary. Each subsystem corresponds to coupled hyperbolic partial differential equations. Such in-domain actuation leads to higher complexity, and represents a significant difference with existing results. Then, starting from a classical controllability condition, we design a state feedback control law for the considered class of systems. The proposed approach is based on the backstepping methodology. However, to deal with the complex structure of the system, we use Fredholm integral transforms instead of classical Volterra transforms. We prove the invertibility of such transforms using an original operator framework. The well posedness of the backstepping kernel equations defining the transformations is also shown with the same arguments. By using a similar procedure, we are then able to design a Luenberger-type observer. Finally, we use the state estimation in the stabilizing controller to obtain an output-feedback law. Some test cases complete the paper.