Approximation of null controls for semilinear heat equations using a least-squares approach

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 27; p. 63
• Main Authors: Lemoine, Jérôme, Marín-Gayte, Irene, Münch, Arnaud
• Format: Journal Article
• Language: English
• Published: EDP Sciences 2021
• Subjects: Mathematics; Optimization and Control; Null controllability; Semilinear heat equation; Least-squares approach
• ISSN: 1292-8119; 1262-3377
• DOI: 10.1051/cocv/2021062

The null distributed controllability of the semilinear heat equation ∂ t y − Δ y + g ( y ) = f 1 ω assuming that g ∈ C 1 (ℝ) satisfies the growth condition lim sup | r |→ ∞ g ( r )∕(| r |ln 3∕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.