Backstepping control for a class of coupled hyperbolic-parabolic PDE systems

In this work, we consider the boundary stabilization of a linear diffusion equation coupled with a linear transport equation. This type of hyperbolic-parabolic partial differential equations (PDEs) coupling arises in many biological, chemical and thermal systems. The two equations are coupled inside the domain and at the boundary. The in-domain coupling architecture is considered from both sides i.e. an advection source term driven by the transport PDE and a Volterra integral source term driven by the parabolic PDE. Using a backstepping method, we derive two feedback control laws and we give sufficient conditions for the exponential stability of the coupled system in the L-2 norm. Controller gains are calculated by solving hyperbolic-parabolic kernel equations arising from the backstepping transformations. The theoretical results are illustrated by numerical simulations.