Stabilization and control for the biharmonic Schrödinger equation

Abstract

The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schrödinger system: $$i{\partial }_{t}u+{\partial }_x^{2}u-{\partial }_x^{4}u=\lambda |u{|}^{2}u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exact controllability ensures that fourth order nonlinear Schrödinger is globally exactly controllable.

24 pages

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