Backstepping control of mixed hyperbolic-parabolic PDE system with multiple coupling terms

This paper considers a class of hyperbolic-parabolic PDE system with mixed-coupling terms, a rather unexplored family of systems. Compared with the previous literature, the coupled system we explore contains more interior-coupling terms, which makes controller design more challenging. Our goal is to design a boundary controller to stabilise the coupled system exponentially. For that, we propose two alter-native controllers whose design is based on the backstepping method; the difference between them is in the target system. The first one keeps one of the coupling terms in the hyperbolic PDE which is compensated by adding an stabilising term in the parabolic PDE; however, the price to pay is that the coefficient of the term might become rather large. The second alternative is more involved but results in a target system which is directly stable, with arbitrary convergence rate. After that, we analyse the stability of the two target systems in the L
²
×H
¹
sense. The (highly coupled) backstepping kernels are derived, and their well-posedness is shown in the appropriate spaces by an infinite induction energy series. Moreover, we show the invertibility of transformations by displaying the inverse transformations; this guarantees closed-loop stability. Finally, numerical simulations are implemented, and the results illustrate that the control laws derived from the two transformations can both stabilise the mixed PDE system exponentially, showing the difference between them; namely, that the first transformation imposes larger terms and therefore larger kernels than the second one to stabilise the controlled system.