Positive and negative exact boundary controllability results for the linear Biharmonic Schrödinger equation

Abstract

In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation $i{\partial }_{t}y=-{\partial }_x^{4}y+\gamma {\partial }_x^{2}y$ on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the {left} endpoint, where the parameter $\gamma <0$. We prove that this system is exactly controllable in time $T>0$, if and only if, the parameter $\gamma$ does not belong to a critical countable set of negative real numbers. The analysis in this work is based on spectral analysis together with the nonharmonic Fourier series method.