  In this paper, we consider a nonlinear system of two parabolic equations,
with a distributed control in the first equation and an odd coupling term in
the second one. We prove that the nonlinear system is small-time locally
null-controllable. The main difficulty is that the linearized system is not
null-controllable. To overcome this obstacle, we extend in a nonlinear setting
the strategy introduced by the first author that consists in constructing odd
controls for the linear heat equation. The proof relies on three main steps.
First, we obtain from the classical L^2 parabolic Carleman estimate, conjugated
with maximal regularity results, a weighted L^p observability inequality for
the nonhomogeneous heat equation. Secondly, we perform a duality argument,
close to the well-known Hilbert Uniqueness Method in a reflexive Banach
setting, to prove that the heat equation perturbed by a source term is
null-controllable thanks to odd controls. Finally, the nonlinearity is handled
with a Schauder fixed-point argument.
