Stabilization of an unstable reaction-diffusion PDE cascaded with a heat equation

We consider a control problem of an unstable reaction-diffusion parabolic PDE cascaded with a heat equation through a boundary, where the heat influx of the heat equation is fed into the temperature of the reaction-diffusion equation, and the control actuator is designed at the another boundary of the heat equation. A backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed-loop system is equivalent to a cascade of PDE-PDE system, which is shown to be exponentially stable in a suitable Hilbert space. With the Dirichlet boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H-¹(0,1). Numerical simulations are presented to illustrate the convergence of the state of the reaction-diffusion equation.