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An explicit output-feedback boundary feedback law is introduced thatstabilizes an unstable linear constant-coefficient reaction-diffusion equationon an $n$-ball (which in 2-D reduces to a disk and in 3-D reduces to a sphere)using only measurements from the boundary. The backstepping method is used todesign both the control law and a boundary observer. To apply backstepping thesystem is reduced to an infinite sequence of 1-D systems using sphericalharmonics. Well-posedness and stability are proved in the $H^1$ space. Theresulting control and output injection gain kernels are the product of thebackstepping kernel used in control of one-dimensional reaction-diffusionequations and a function closely related to the Poisson kernel in the $n$-ball.

Boundary Control of Reaction-Diffusion PDEs on Balls in Spaces of Arbitrary Dimensions