Boundary feedback stabilization of a Schrödinger equation interconnected with a heat equation

In this paper, we study stabilization for a Schrödinger equation, which is interconnected with a heat equation via boundary coupling. A direct boundary feedback control is adopted. By a detailed spectral analysis, it is found that there are two branches of eigenvalues: one is along the negative real axis, and the other is approaching to a vertical line, which is parallel to the imagine axis. Moreover, it is shown that there is a set of generalized eigenfunctions, which forms a Riesz basis for the energy state space. Finally, the spectrum-determined growth condition is held and the exponential stability of the system is then concluded.