Backstepping control design for a coupled hyperbolic-parabolic mixed class PDE system

We consider the class of linear partial differential equation systems in 1-D involving two distinct families, hyperbolic and parabolic. The model is motivated by physical phenomena arising in extreme ultraviolet light generation (EUV) for next-generation photolithography, where the droplet transport stream is affected by the diffusion of plasma generated from droplet vaporization. Other systems that can exhibit this behavior are found in biological chemotaxis and simplified predator-prey models. The two PDEs are coupled in-domain, which brings additional complexity when compared with the boundary coupled case, which has been explored in previous literature by Krstic (2009). The coupling is considered in an unidirectional manner, where the controlled hyperbolic PDE modeling the droplet stream displacement exhibits an in-domain Volterra integral source driven by the parabolic PDE modeling the plasma diffusion, which itself is driven at the boundary via the hyperbolic PDE. A backstepping method is derived to transform the system into a chosen exponentially stable target system, in the L² × H¹ norm. The backstepping method gives rise to a coupled hyperbolic-parabolic gain kernel PDE which must be solved to determine the controller. Simulation studies are presented to illustrate the effectiveness of the control.