Bilateral backstepping control of coupled linear parabolic PDEs with spatially varying coefficients

TL;DR

This paper develops a backstepping control method for coupled linear parabolic PDEs with spatially varying coefficients, using folding techniques and integral equations to achieve exponential stabilization with bilateral actuation.

Contribution

It introduces a novel systematic approach to solve kernel equations for coupled PDEs with spatially varying coefficients using folding and integral equations.

Findings

Successfully stabilizes coupled parabolic PDEs with bilateral actuation.

Provides a systematic method to solve complex kernel equations.

Demonstrates exponential stability with a tunable decay rate.

Abstract

This paper considers the backstepping state feedback control of coupled linear parabolic PDEs with spatially varying coefficients and bilateral actuation. By making use of the folding technique, a system representation with unilateral actuation is obtained, allowing to apply the standard backstepping transformation. To ensure the regularity of the solution, the folded system is subject to unusual folding boundary conditions, which lead to additional boundary couplings between the PDEs. Therefore, the solution of the corresponding kernel equations determining the transformations is a very challenging problem. A systematic approach to derive the corresponding integral equations is proposed, allowing to solve them with the method of successive approximations. By making use of a Volterra and a Volterra-Fredholm transformation, the closed-loop system is mapped into a cascade of stable parabolic systems. This allows a simple proof of exponential stability in the $L_2$-norm with the decay rate as design parameter. The bilateral state feedback stabilization of an unstable system of two coupled parabolic PDEs and the comparison to the application of an unilateral controller demonstrates the results of the paper.