  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.
