Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls (Q2835352)

The authors introduce an explicit output-feedback 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) using only measurements from the boundary. The backstepping method is used to design both the control law and a boundary observer. To apply backstepping the system is reduced to an infinite sequence of \(1\)-D systems using spherical harmonics. Well-posedness and stability are proved in the \(L^2\) and \(H^1\) spaces. The resulting control and output injection gain kernels are 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.