Controller design to stabilization of Schrödinger equation with boundary input disturbance

ABSTRACT

In this paper, the stabilization problem of one-dimensional Schrödinger equation with boundary disturbance is concerned. The variable structure control technique is adopted to design the nonlinear feedback controller. For the Schrödinger system, the observation blind point problem is discussed and the relationship between the initial data and observation blind point is given. According to a theorem which can be regarded as an extended version of the Lions–Lax–Milgram theorem, the existence and uniqueness of the solution for the system are proved. Finally, a sufficient condition for the asymptotic stability of the system is provided. As a remark, the asymptotic stability, which can be realized by the control, is also presented.

COMMUNICATED BY:

• Daomin Cao

KEYWORDS:

• Schrödinger equation
• nonlinear boundary control
• disturbance
• observation blind point
• asymptotic stability

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 93D15
• 93D20
• 35B40

Acknowledgments

The authors would like to thank the anonymous referees and Editor in Chief for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Genqi Xu  http://orcid.org/0000-0001-8120-0490

Additional information

Funding

This work was supported by the Natural Science Foundation of China [grant numbers NSFC-61773277, 61573252].