Boundary control of a singular reaction-diffusion equation on a disk

TL;DR

This paper extends boundary stabilization techniques to a 2-D reaction-diffusion equation with radially-varying coefficients on a disk, addressing singularities in the backstepping kernel equations using Catalan numbers.

Contribution

It introduces a novel approach to prove well-posedness of singular kernel equations in backstepping for radially-varying coefficients on a disk.

Findings

Successfully stabilizes reaction-diffusion equations with spatially-varying coefficients

Develops a non-standard proof using Catalan numbers for singular integral equations

Extends backstepping method to more complex geometries with singularities

Abstract

Recently, the problem of boundary stabilization for unstable linear constant-coefficient reaction-diffusion equation on N-balls has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. This work deals with radially-varying reaction coefficients under revolution symmetry conditions on a disk (the 2-D case). Under these conditions, the equations become singular in the radius. When applying the backstepping method, the same type of singularity appears in the backstepping kernel equations. Traditionally, well-posedness of the kernel equations is proved by transforming them into integral equations and then applying the method of successive approximations. In this case, the resulting integral equation is singular. A successive approximation series can still be formulated, however its convergence is challenging to show due to the singularities. The problem is solved by a rather non-standard proof that uses the properties of the Catalan numbers, a well-known sequence frequently used in combinatorial mathematics.