Homogeneity applied to the controllability of a system of parabolic equations

We consider a system of two parabolic equations with a forcing term present in one equation and a nonlinear coupling term of a particular type in the other one. We establish a local null controllability result, even if the linearized control system fails to be null controllable. The proof combines the return method, a Carleman inequality and an argument based on homogeneity. Our result can be applied to a reaction-diffusion system from Chemistry and to the Ginzburg-Landau equation.