Carleman estimates and exact boundary controllabilty for a system of coupled non-conservative Schödinger Equations

Abstract

We consider a system of 2 (or more) coupled Schródinger equations in the difficult situation where the equations have first-order, lower-order terms, as well as first-order coupling in all space variables. By using a general differential multiplier we give a `friendly’ proof of earleman estimates. Under more restrictive intrinsic conditions mostly on the coupling operators, we obtain eaact controllability results for the coupled system, under various combinations of boundary controls: Dirichlet/Dirichlet; Dirichlet/Neumann; Neumann/Neumann. The controls are active on a suitable portion of the boundary. These results cannot be obtained by standard multipliers.