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

Abstract

We consider the semilinear heat equation posed on a smooth bounded domain Ω of RN with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset ω of Ω. The goal of this paper is to prove the uniform large time global null-controllability for semilinearities f(s)=±|s|logα⁡(1+|s|) where α∈[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⁡(C‖a‖∞1/2) for nonnegative initial data. This inequality comes from a new L1 Carleman estimate. A Kakutani-Leray-Schauder'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∞(Ω)-norm, and the local null-controllability of the semilinear heat equation.