Finite-dimensional homogeneous boundary control for a 1D reaction-diffusion equation

In this paper, we address the problem of finite-dimensional boundary stabilization of the 1 D reaction-diffusion equation. Using the modal decomposition approach, we propose a finite-dimensional homogeneous controller, stabilizing the unstable dynamics while ensuring the stability of the residual part. The closed-loop system with homogeneous feedback is well-posed and stable. The proposed controller is proven superior to a finite-dimensional linear feedback controller in terms of closed-loop performance. The numerical simulations are presented to support the analytical results.