Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls
1 Department of Aerospace Engineering, Universidad de Sevilla,
Camino de los Descubrimiento s.n., 41092 Sevilla, Spain.
2 Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411, USA.
Accepted: 7 June 2016
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 L2 and H1 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.
Mathematics Subject Classification: 35K57 / 93D20 / 93D30 / 33C55
Key words: Infinite-dimensional backstepping / boundary control / boundary observer / reaction-diffusion system / spherical harmonics
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