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Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2020

Global null-controllability and nonnegative-controllability of slightly superlinear heat equations

Résumé

We consider the semilinear heat equation posed on a smooth bounded domain $\Omega$ of $\mathbb{R}^{N}$ with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset $\omega$ of $\Omega$. The goal of this paper is to prove the uniform large time global null-controllability for semilinearities $f(s) = \pm |s| \log^{\alpha}(2+|s|)$ where $\alpha \in [3/2,2)$ which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global nonnegative-controllability (respectively nonpositive-controllability) of the system, i.e., one can steer any initial data to a nonnegative (respectively nonpositive) state in arbitrary time. In particular, one can act locally thanks to the control term in order to prevent the blow-up from happening. The proof relies on precise observability estimates for the linear heat equation with a bounded potential $a(t,x)$. More precisely, we show that observability holds with a sharp constant of the order $\exp\left(C |a|_{\infty}^{1/2}\right)$ for nonnegative initial data. This inequality comes from a new $L^1$ Carleman estimate. A Kakutani's fixed point argument enables to go back to the semilinear heat equation. Secondly, the uniform large time null-controllability result comes from three ingredients: the global nonnegative-controllability, a comparison principle between the free solution and the solution to the underlying ordinary differential equation which provides the convergence of the free solution toward $0$ in $L^{\infty}(\Omega)$-norm, and the local null-controllability of the semilinear heat equation.
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Dates et versions

hal-01907005 , version 1 (28-10-2018)
hal-01907005 , version 2 (30-10-2018)

Identifiants

Citer

Kévin Le Balc’h. Global null-controllability and nonnegative-controllability of slightly superlinear heat equations. Journal de Mathématiques Pures et Appliquées, 2020, 135, pp.103-139. ⟨10.1016/j.matpur.2019.10.009⟩. ⟨hal-01907005v2⟩
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