Weakly-damped focusing nonlinear Schrödinger equations with Dirichlet control

In this article we consider the weakly damped focusing nonlinear Schrödinger equations on bounded domains at the natural H1-energy level with Dirichlet control acting on a portion of the boundary. We introduce the dynamic extension method for homogenizing the inhomogeneous boundary input. Then, we construct approximate solutions using monotone operator theory. A hidden trace regularity is proved to control the norm of the solutions in a global sense. This allows the use of compactness techniques by which we prove the existence of weak solutions. Finally, using multiplier techniques, we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion.