Finite dimensional backstepping controller design

TL;DR

This paper proposes a finite-dimensional backstepping controller for PDE stabilization that uses only a finite number of Fourier modes, simplifying implementation while maintaining effectiveness.

Contribution

It introduces a novel finite-dimensional backstepping method applicable to PDEs, with theoretical analysis and numerical validation for mode selection and stabilization rates.

Findings

Effective stabilization with fewer Fourier modes

Quantitative estimates for mode number needed for desired decay rates

Numerical results confirm theoretical predictions

Abstract

We introduce a finite dimensional version of backstepping controller design for stabilizing solutions of PDEs from boundary. Our controller uses only a finite number of Fourier modes of the state of solution, as opposed to the classical backstepping controller which uses all (infinitely many) modes. We apply our method to the reaction-diffusion equation, which serves only as a canonical example but the method is applicable also to other PDEs whose solutions can be decomposed into a slow finite-dimensional part and a fast tail, where the former dominates the evolution in large time. One of the main goals is to estimate the sufficient number of modes needed to stabilize the plant at a prescribed rate. In addition, we find the minimal number of modes that guarantee the stabilization at a certain (unprescribed) decay rate. Theoretical findings are supported with numerical solutions.