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

Kuntal Bhandari (1) , Jérôme Lemoine (2, 3) , Arnaud Munch (4, 2)

1 IM / CAS - Institute of Mathematics of the Czech Academy of Science
2 UCA - Université Clermont Auvergne
3 Laboratoire de Mathématiques Blaise Pascal (LMBP)
4 LMBP - Laboratoire de Mathématiques Blaise Pascal

This paper addresses the boundary null-controllability of the semi-linear heat equation ∂ty − ∂xxy + f (y) = 0, (x, t) ∈ (0, 1) × (0, T). Assuming that the nonlinear function f is locally Lipschitz and satisfies lim sup |r|→+∞ |f (r)|/(|r| ln 3/2 |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 is C 1 over R and satisfies lim sup |r|→∞ |f (r)|/ ln 3/2 |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 L 2 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.

Mots clés: Mathematics Subject Classification: 35K58 93B05 Semilinear heat equation Boundary null-controllability Carleman estimates Fixed point arguments; Mathematics Subject Classification: 35K58; 93B05 Semilinear heat equation; Boundary null-controllability; Carleman estimates; Fixed point arguments

Domaines: Mathématiques [math]

Kuntal Bhandari, Jérôme Lemoine, Arnaud Munch. Global boundary null-controllability of one-dimensional semilinear heat equations. 2023. ⟨hal-03938241⟩