Null-Controllability of Two Species Reaction-Diffusion System with Nonlinear Coupling: A New Duality Method

We consider a $2\times2$ nonlinear reaction-diffusion system posed on a smooth bounded domain$$\Omega$of R N (N$\ge$1$). The control input is in the source term of only one equation. It is localized in some arbitrary nonempty open subset $\omega$ of the domain $\Omega$. First, we prove a global null-controllability result when the coupling term in the second equation is an odd power. As the linearized system around zero is not null-controllable, the usual strategy consists in using the return method, introduced by Jean-Michel Coron, or the method of power series expansions. In this paper, we give a direct nonlinear proof, which relies on a new duality method that we call Reflexive Uniqueness Method. It is a variation in reflexive Banach spaces of the well-known Hilbert Uniqueness Method, introduced by Jacques-Louis Lions. It is based on Carleman estimates in $L^p$ ($2 \leq p < \infty$) obtained from the usual Carleman inequality in $L^2$ and parabolic regularity arguments. This strategy enables us to find a control of the heat equation, which is an odd power of a regular function. Another advantage of the method is to produce small controls for small initial data. Secondly, thanks to the return method, we also prove a null-controllability result for more general nonlinear reaction-diffusion systems, where the coupling term in the second equation behaves as an odd power at zero.