Bilateral boundary control of an input delayed 2-D reaction-diffusion equation

TL;DR

This paper develops a PDE backstepping-based delay compensation method for bilateral input delays in a 2-D reaction-diffusion PDE, introducing novel transformations and addressing singular kernel equations for improved control stability.

Contribution

It introduces a new backstepping approach with singular kernel solutions for 2-D PDEs with bilateral delays, expanding delay compensation techniques to higher-dimensional systems.

Findings

The proposed method effectively compensates for bilateral input delays.

Kernel equations are solved as double trigonometric series with singularities.

Numerical simulations confirm the stability and effectiveness of the control design.

Abstract

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction-diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy-Schwarz inequality, the 2-D Fourier series, and the Parseval's theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.