Electron. J. Diff. Equ., Vol. 2016 (2016), No. 216, pp. 1-17.

Well-posedness and exact controllability of a fourth order Schrodinger equation with variable coefficients and Neumann boundary control and collocated observation

Ruili Wen, Shugen Chai

Abstract:
We consider an open-loop system of a fourth order Schrodinger equation with variable coefficients and Neumann boundary control and collocated observation. Using the multiplier method on Riemannian manifold we show that that the system is well-posed in the sense of Salamon. This implies that the exponential stability of the closed-loop system under the direct proportional output feedback control and the exact controllability of open-loop system are equivalent. So in order to conclude feedback stabilization from well-posedness, we study the exact controllability under a uniqueness assumption by presenting the observability inequality for the dual system. In addition, we show that the system is regular in the sense of Weiss, and that the feedthrough operator is zero.

Submitted May 25, 2016. Published August 12, 2016.
Math Subject Classifications: 93C20, 35L35, 35B37.
Key Words: Fourth order Schrodinger equation; variable coefficients; well-posedness; exact controllability; boundary control; boundary observation.

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Ruili Wen
School of Mathematical Sciences
Shanxi University, Taiyuan 030006, China
email: wenruili@sxu.edu.cn
Shugen Chai
School of Mathematical Sciences
Shanxi University, Taiyuan 030006, China
email: sgchai@sxu.edu.cn

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