Backstepping Control of an Underactuated Hyperbolic–Parabolic Coupled PDE System

This article considers a class of hyperbolic–parabolic partial differential equation (PDE) system with some interior mixed-coupling terms, a rather unexplored family of systems. The family of systems we explore contains several interior-coupling terms, which makes controller design more challenging. Our goal is to design a boundary controller to exponentially stabilize the coupled system. For that, we propose a controller whose design is based on the backstepping method. Under this controller, we analyze the stability of the closed loop in the
$H^{1}$
sense. A set of (highly coupled) backstepping kernel equations is derived, and their well-posedness is shown in the appropriate spaces by an infinite induction energy series, which has not been used before in this setting. Moreover, we show the invertibility of transformations by displaying the inverse transformations, as required for closed-loop well-posedness and stability. Finally, a numerical simulation is implemented, and the result illustrates that the control law designed by the backstepping transformation can stabilize the mixed PDE system exponentially.