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)H−1(0,1). Numerical simulations are presented to illustrate the convergence of the state of the reaction–diffusion equation.