Backstepping Control for a Class of Hyperbolic-Parabolic Systems

We consider a class of mixed-coupling hyperbolic-parabolic partial differential equation (PDE) systems, motivated by the melt spinning process in chemical fiber industry. This system involves a new type of coupling, where the derivative term of the hyperbolic state is coupled into the parabolic equation. This is a distinction from existing studies, which primarily consider source term or integral term coupling. In the paper, we propose a two-step backstepping transformation and design two boundary controllers: one for the parabolic PDE and the other for the hyperbolic PDE. We prove the well-posedness of the coupled kernel equations using an infinite induction energy series approach. Additionally, we demonstrate the invertibility of the transformation and establish the exponential stability of the closed-loop system in the L 2 norm. Finally, the theoretical results are validated through numerical simulations.