Geodesic
Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls
Vazquez, Rafael1  ; Krstic, Miroslav2 
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.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1078-1096

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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.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016033
Classification : 35K57, 93D20, 93D30, 33C55
Keywords: Infinite-dimensional backstepping, boundary control, boundary observer, reaction-diffusion system, spherical harmonics

Vazquez, Rafael 1 ; Krstic, Miroslav 2

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. 
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     author = {Vazquez, Rafael and Krstic, Miroslav},
     title = {Explicit output-feedback boundary control of reaction-diffusion {PDEs} on arbitrary-dimensional balls},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1078--1096},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {4},
     year = {2016},
     doi = {10.1051/cocv/2016033},
     mrnumber = {3570495},
     zbl = {1358.35058},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2016033/}
}
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Vazquez, Rafael; Krstic, Miroslav. Explicit output-feedback boundary control of reaction-diffusion PDEs on arbitrary-dimensional balls. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1078-1096. doi: 10.1051/cocv/2016033

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