On the Controllability of a Parabolic–Elliptic System Governed by a Bilinear Control

This paper deals with the analysis of the internal bilinear control for two nonlinear parabolic–elliptic systems. In a first case, we study the exact controllability to the trajectories for the system in which the control function acts on the parabolic equation. In the second case, when the control function acts on the elliptic equation, we control to the trajectories the parabolic component while keeping the elliptic component uniformly bounded. Our approach, in both cases, relies on an application of Liusternik's inverse mapping theorem which demands the proof of suitable Carleman estimate.