Control of a 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 other 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 target system of PDE-PDE cascades, which is shown to be exponentially stable in some Hilbert space. With the boundary input from the heat equation, the reaction-diffusion PDE is shown to be exponentially stable in H
⁻¹
(0, 1), and the stability of the heat equation is shown to be characterized in terms of a subspace of H
¹
(0, 1) rather than the usual L
²
(0, 1).