The main goal of this article is to study computationally the controllability of a diffusion process on the surface of a sphere in . To achieve this goal, we employ a methodology combining finite differences for the time discretization, finite elements for the space approximation, and a conjugate gradient algorithm for the iterative solution of the discrete control problems. The results of numerical experiments, obtained using the above methodology, will be presented. Furthermore, the null-controllability properties of the diffusion model under consideration will be also studied computationally.
Accepté le :
DOI : 10.1051/cocv/2016045
Mots clés : Diffusion process, surface of a shere, conjugate gradient, null-controlability, approximate controllability, Laplace−Beltrami operator
@article{COCV_2016__22_4_1054_0,
author = {Assaely Le\'on Velasco, D. and Glowinski, Roland and H\'ector Ju\'arez Valencia, L.},
title = {On the controllability of diffusion processes on a sphere: {A} numerical study},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1054--1077},
publisher = {EDP-Sciences},
volume = {22},
number = {4},
year = {2016},
doi = {10.1051/cocv/2016045},
mrnumber = {3570494},
zbl = {1353.49042},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016045/}
}
TY - JOUR AU - Assaely León Velasco, D. AU - Glowinski, Roland AU - Héctor Juárez Valencia, L. TI - On the controllability of diffusion processes on a sphere: A numerical study JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1054 EP - 1077 VL - 22 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016045/ DO - 10.1051/cocv/2016045 LA - en ID - COCV_2016__22_4_1054_0 ER -
%0 Journal Article %A Assaely León Velasco, D. %A Glowinski, Roland %A Héctor Juárez Valencia, L. %T On the controllability of diffusion processes on a sphere: A numerical study %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1054-1077 %V 22 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016045/ %R 10.1051/cocv/2016045 %G en %F COCV_2016__22_4_1054_0
Assaely León Velasco, D.; Glowinski, Roland; Héctor Juárez Valencia, L. On the controllability of diffusion processes on a sphere: A numerical study. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1054-1077. doi : 10.1051/cocv/2016045. http://www.numdam.org/articles/10.1051/cocv/2016045/
and , On the nodal set of the eigenfunctions of the Laplace−Beltrami operator for bounded surfaces in ; A computational approach. Commun. Pure Appl. Anal. 13 (2014) 2115–2126. | DOI | MR | Zbl
and , Convergence analysis of variational and non–variational multigrid algorithms for the Laplace−Beltrami operator. Math. Comput. 81 (2012) 1263–1288. | DOI | MR | Zbl
, and , On exact and approximate boundary controllabilities for the heat equation: A numerical approach. J. Optim. Theory Appl. 82 (1994) 429–484. | DOI | MR | Zbl
and , Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429–448. | MR | Zbl
and , Strong convergent approximations of null controls for the 1D heat equation. SEMA Journal 61 (2013) 49–78. | DOI | MR | Zbl
and , Numerical null controllability of the 1D heat equation: Carleman weights and duality. J. Optim. Theory Appl. 163 (2014) 253–285. | DOI | MR | Zbl
R. Glowinski, Numerical Methods for Nonlinear Variational Problems, printing (2008). Springer, New York (1984). | MR | Zbl
R. Glowinski, Finite element methods for incompressible viscous flow. Vol. 9 of Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions. North-Holland, Amsterdam (2003) 3–1176. | MR | Zbl
R. Glowinski and D.C. Sorensen, Computing the eigenvalues of the Laplace−Beltrami operator on the surface of a torus: A numerical approach. In Partial Differential Equations: Modelling and Numerical Simulation. Springer Netherlands (2008) 225–232. | MR | Zbl
R. Glowinski, J.L. Lions and J.W. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Cambridge University Press, Cambridge, UK (2008). | Zbl
and , Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. | DOI | MR | Zbl
, and , On the controllability of diffusion processes on the surface of a torus: A computational approach. Pacific J. Optim. 11 (2015) 763–790. | MR | Zbl
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York (1971). | MR | Zbl
, Geometric bounds on the grow rate of null-controllability cost for the heat equation in small time. J. Differ. Equ. 204 (2004) 202–226. | DOI | MR | Zbl
, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds. Math. Res. Lett. 12 (2005) 37–47. | DOI | MR | Zbl
and , Numerical approximation of null controls for the heat equation: Ill-posedness and remedies. Inverse Problems 26 (2010) 085018. | DOI | MR | Zbl
and , Numerical null controllability of the heat equation through a least squares and variational approach. Eur. J. Appl. Math. 25 (2014) 277–306. | DOI | MR | Zbl
, and , Optimal shape and location of sensors for parabolic equations with random initial data. Arch. Rational Mech. Anal. 216 (2015) 921–981. | DOI | MR | Zbl
E. Zuazua, Control and numerical approximation of the wave and heat equations. In Vol. 3 of Proc. Internat. Congress Math. Madrid, Spain (2006) 1389–1417. | MR | Zbl
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