Global boundary null-controllability of one-dimensional semilinear heat equations

0586683 - MÚ 2025 RIV US eng J - Journal Article
Bhandari, Kuntal - Lemoine, J. - Münch, A.
Global boundary null-controllability of one-dimensional semilinear heat equations.
Discrete and Continuous Dynamical systems - Series S. Roč. 17, č. 7 (2024), s. 2251-2297. ISSN 1937-1632. E-ISSN 1937-1179
Institutional support: RVO:67985840
Keywords : boundary null-controllability * Carleman estimates * fixed point arguments
OECD category: Pure mathematics
Impact factor: 1.8, year: 2022
Method of publishing: Open access
https://doi.org/10.3934/dcdss.2024003

This paper addresses the boundary null-controllability of the semilinear heat equation ∂ty - ∂xxy + f(y) = 0, (x, t) ∈ (0, 1) × (0, T). Assuming that the function f ∈ C1(R) satisfies lim sup|r|→+∞ |f(r)|/(|r| ln3/2 |r|) ≤ β for some β > 0 small enough and that the initial datum belongs to L∞(0, 1), we prove the global null-controllability using the Schauder fixed point theorem and a linearization for which the term f(y) is seen as a right side of the equation. Then, assuming that f satisfies lim sup|r|→∞ |f'(r)|/ ln3/2 |r| ≤ β for some β small enough, we show that the fixed point application is contracting yielding a constructive method to approximate boundary controls for the semilinear equation. The crucial technical point is a regularity property of a state-control pair for a linear heat equation with L2 right hand side obtained by using a global Carleman estimate with boundary observation. Numerical experiments illustrate the results. The arguments developed can notably be extended to the multi-dimensional case.
Permanent Link: https://hdl.handle.net/11104/0354115