The null distributed controllability of the semilinear heat equation ∂$$y − Δy + g(y) = f 1$$ assuming that g ∈ C1(ℝ) satisfies the growth condition lim sup$$g(r)∕(|r|ln3∕2|r|) = 0 has been obtained by Fernández-Cara and Zuazua (2000). The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming that g′ is bounded and uniformly Hölder continuous on ℝ with exponent p ∈ (0, 1], we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order 1 + p after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton method: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.
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Keywords: Semilinear heat equation, null controllability, least-squares method
@article{COCV_2021__27_1_A65_0,
author = {Lemoine, J\'er\^ome and Mar{\'\i}n-Gayte, Irene and M\"unch, Arnaud},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Approximation of null controls for semilinear heat equations using a least-squares approach},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021062},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021062/}
}
TY - JOUR AU - Lemoine, Jérôme AU - Marín-Gayte, Irene AU - Münch, Arnaud ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Approximation of null controls for semilinear heat equations using a least-squares approach JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021062/ DO - 10.1051/cocv/2021062 LA - en ID - COCV_2021__27_1_A65_0 ER -
%0 Journal Article %A Lemoine, Jérôme %A Marín-Gayte, Irene %A Münch, Arnaud %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Approximation of null controls for semilinear heat equations using a least-squares approach %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021062/ %R 10.1051/cocv/2021062 %G en %F COCV_2021__27_1_A65_0
Lemoine, Jérôme; Marín-Gayte, Irene; Münch, Arnaud. Approximation of null controls for semilinear heat equations using a least-squares approach. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 63. doi: 10.1051/cocv/2021062
[1] , Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73–89.
[2] , and , Constructive exact control of semilinear multi-dimensional wave equations. Preprint. | arXiv
[3] , On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems, in CANUM 2012, 41e Congrès National d’Analyse Numérique, volume 41 of ESAIM Proc.. EDP Sci., Les Ulis (2013) 15–58.
[4] and Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014) 1035–1078.
[5] , and , On exact and approximate boundary controllabilities for the heat equation: a numerical approach. J. Optim. Theory Appl. 82 (1994) 429–484.
[6] and , An introduction to semilinear evolution equations, Volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998). Translated from the 1990 French original by Yvan Martel and revised by the authors.
[7] , The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
[8] , Control and nonlinearity, volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007).
[9] and , Global steady-state controllability of one-dimensional semilinear heat equations. SIAM J. Control Optim. 43 (2004) 549–569.
[10] , Newton methods for nonlinear problems. Vol. 35 of Springer Series in Computational Mathematics. Springer, Heidelberg (2011). Affine invariance and adaptive algorithms, First softcover printing of the 2006 corrected printing.
[11] , and , On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 1–41.
[12] and , Global Carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control Optim. 45 (2006) 1399–1446.
[13] and , Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods. Math. Control Relat. Fields 2 (2012) 217–246.
[14] and , Strong convergent approximations of null controls for the 1D heat equation. SeMA J. 61 (2013) 49–78.
[15] and , Numerical exact controllability of the 1D heat equation: duality and Carleman weights. J. Optim. Theory Appl. 163 (2014) 253–285.
[16] and , Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 583–616.
[17] and . Imanuvilov, Controllability of evolution equations, volume 34 of Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996).
[18] , New development in Freefem++. J. Numer. Math. 20 (2012) 251–265.
[19] and , Uniform controllability of semidiscrete approximations of parabolic control systems. Systems Control Lett. 55 (2006) 597–609.
[20] Global null-controllability and nonnegative-controllability of slightly superlinear heat equations. J. Math. Pures Appl. 135 (2020) 103–139.
[21] and , Constructive exact controls for semilinear one-dimensional heat equations. Preprint. | arXiv
[22] and , A fully space-time least-squares method for the unsteady Navier-Stokes system. Preprint. In revision in J. of Mathematical Fluid Mechanics. | arXiv
[23] and , Resolution of the implicit Euler scheme for the Navier-Stokes equation through a least-squares method. Numer. Math. 147 (2021) 349–391.
[24] , and , Analysis of continuous H−1-least-squares approaches for the steady Navier-Stokes system. Appl. Math. Optim. 83 (2021) 461–488.
[25] , A least-squares formulation for the approximation of controls for the Stokes system. Math. Control Signals Syst. 27 (2015) 49–75.
[26] and , Numerical null controllability of the heat equation through a least squares and variational approach. Eur. J. Appl. Math. 25 (2014) 277–306.
[27] and , A mixed formulation for the direct approximation of L2 -weighted controls for the linear heat equation. Adv. Comput. Math. 42 (2016) 85–125.
[28] and , Constructive exact control of semilinear 1d wave equations by a least-squares approach. Preprint. | arXiv
[29] and , Numerical approximation of null controls for the heat equation: ill-posedness and remedies. Inverse Probl. 26 (2010) 085018.
[30] , A damped Newton algorithm for computing viscoplastic fluid flows. J. Non-Newton. Fluid Mech. 238 (2016) 6–15.
[31] , Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 109–129.
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