Stabilization by 1D Boundary Actuation of Distal 1D Reaction-Diffusion PDE through Heat PDE on a Rectangle

This paper presents a backstepping control design method of stabilization unstable 1D reaction-diffusion system, where the input is a 1D function on an edge of a rectangle, the distal system is a 1D reaction-diffusion PDE on the opposite edge of the rectangle, and the actuator dynamics in between are a 2D heat PDE on the rectangle between the opposite edges. A novel invertible integral transformation is introduced and the resulting controller with feedback of both PDEs' states (the distal 1D and the interior 2D states). We define a new Lyapunov function that contains cosine coefficients to prove the exponential stability in H-2 norm of the closed-loop system. Finally, the theoretical result is illustrated by simulations on a numerical example.