Bilinear control for the parabolic-elliptic system

In the present paper we study the global approximate controllability of the parabolic-elliptic system, which has caught a great deal of attention,governed in a bounded interval by a pair of controls: (i)the traditional internal either locally distributed or lumped control and (ii) the lumped control entering the system as a time-dependent coefficient. The motivation for the latter is due to the well-known lack of global null-controllability properties for the class of systems governed by the parabolic partial differential equations when they are only steered by the former controls. However, only approximate controllability can be obtained because of the nonlinearity of the parabolic-elliptic system. Our approach involves an asymptotic technique allowing us to separate and combine the impacts generated by the above-mentioned two types of controls.