Explicit boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls

This paper introduces an explicit full-state boundary feedback law that stabilizes an unstable linear constant-coefficient reaction-diffusion equation on an n-ball (which in 2-D reduces to a disk and in 3-D reduces to a sphere). The backstepping method is used to design the control law. To apply backstepping the system is reduced to an infinite sequence of 1-D systems using spherical harmonics. L² well-posedness and stability are proved. The resulting control law is written as a multiple integral whose kernel is the product of the backstepping kernel used in control of one-dimensional reaction-diffusion equations and a function closely related to the Poisson kernel in the n-ball.