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

In this paper, we study the exact boundary controllability of the linear Biharmonic Schrödinger equation $i\partial_ty=-\partial_x^4y+ γ\partial_x^2y$ on a bounded domain with hinged boundary conditions and boundary control acts on the second spatial derivative at the {left} endpoint, where the parameter $γ<0$. We prove that this system is exactly controllable in time $T>0$, if and only if, the parameter $γ$ 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.

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