Approximation of null controls for semilinear heat equations using a least-squares approach
| Autor: | Arnaud Münch, Jérôme Lemoine, Irene Marín-Gayte |
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| Přispěvatelé: | Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), Departamento de Ecuaciones Differenciales y Analysis numérico [Sevilla] (EDAN), Facultad de Matemáticas, Géosciences Environnement Toulouse (GET), Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Observatoire Midi-Pyrénées (OMP), Météo France-Centre National d'Études Spatiales [Toulouse] (CNES)-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD)-Météo France-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD), Institut de Recherche pour le Développement (IRD)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut national des sciences de l'Univers (INSU - CNRS)-Observatoire Midi-Pyrénées (OMP), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS)-Météo-France -Institut de Recherche pour le Développement (IRD)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National d'Études Spatiales [Toulouse] (CNES)-Centre National de la Recherche Scientifique (CNRS)-Météo-France -Centre National de la Recherche Scientifique (CNRS), Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
0209 industrial biotechnology
Pure mathematics Least-squares approach Control and Optimization Constructive proof Mathematics::Analysis of PDEs Fixed-point theorem 93E24 Keywords: Semilinear heat equation 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences 020901 industrial engineering & automation Semilinear heat equation FOS: Mathematics AMS Classifications: 35Q30 Mathematics - Numerical Analysis 0101 mathematics [MATH]Mathematics [math] Finite set Mathematics - Optimization and Control Mathematics 1991 Mathematics Subject Classification.35K58 93B05 93E24 Null controllability Null (mathematics) Order (ring theory) Numerical Analysis (math.NA) least-squares method Computational Mathematics Control and Systems Engineering Optimization and Control (math.OC) Bounded function Exponent Heat equation 35Q30 93E24 [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] |
| Zdroj: | ESAIM: Control, Optimisation and Calculus of Variations ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2021, 27, pp.63. ⟨10.1051/cocv/2021062⟩ ESAIM: Control, Optimisation and Calculus of Variations, 2021, 27, pp.63. ⟨10.1051/cocv/2021062⟩ ESAIM: Control, Optimisation and Calculus of Variations, In press ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, inPress ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, In press |
| ISSN: | 1292-8119 1262-3377 |
| DOI: | 10.1051/cocv/2021062⟩ |
| Popis: | The null distributed controllability of the semilinear heat equation ∂ty − Δy + g(y) = f 1ω assuming that g ∈ C1(ℝ) satisfies the growth condition lim sup|r|→∞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. |
| Databáze: | OpenAIRE |
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