NULL-CONTROLLABILITY OF TWO SPECIES REACTION-DIFFUSION SYSTEM WITH NONLINEAR COUPLING: A NEW DUALITY METHOD

We consider a 2 x 2 nonlinear reaction-diffusion system posed on a smooth bounded domain Omega of R-N (N >= 1). The control input is in the source term of only one equation. It is localized in some arbitrary nonempty open subset w of the domain Omega. First, we prove a global null-controllability result in arbitrary time T > 0 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 of using the return method, introduced by J.-M. 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 the Reflexive Uniqueness Method. It is a variation in reflexive Banach spaces of the well-known Hilbert Uniqueness Method, introduced by J.-L. Lions. It is based on Carleman estimates in L-p (2 <= p < infinity) 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 its ability to produce small controls for small initial data. Second, thanks to the return method, we also prove a local 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.