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

Vazquez, Rafael; Krstic, Miroslav

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1078-1096.

This paper introduces 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.

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