Controllability results for some nonlinear coupled parabolic systems by one control force

In this paper, we present new controllability results for some nonlinear coupled parabolic systems considered in a bounded domain Omega of R-N ( with (with N >= 1 being arbitrary) when the control force acts on a unique equation of the system through an arbitrarily small open set omega subset of Omega. As a model example, we consider a nonlinear phase field system with certain superlinear nonlinearities and prove the null controllability, the exact controllability to the trajectories and the approximate controllability of the model. The crucial point in this paper is the new strategy developed to deal with the null controllability of linear coupled parabolic systems by a unique control force. Global Carleman estimates and the parabolic regularizing effect of the problem are used.