TY - JOUR

T1 - Stabilization and gevrey regularity of a schrdinger equation in boundary feedback with a heat equation

AU - Wang, Jun Min

AU - Ren, Beibei

AU - Krstic, Miroslav

N1 - Funding Information:
Manuscript received December 23, 2010; revised April 11, 2011 and June 08, 2011; accepted July 08, 2011. Date of publication August 12, 2011; date of current version December 29, 2011. This work was supported by the National Natural Science Foundation of China and the Program for New Century Excellent Talents in University of China. Recommended by Associate Editor K. Morris.

PY - 2012/1

Y1 - 2012/1

N2 - We study stability of a Schrdinger equation with a collocated boundary feedback compensator in the form of a heat equation with a collocated input/output pair. Remarkably, exponential stability is achieved for both positive and negative gains, namely, as long as the gain is non-zero. We show that the spectrum of the closed-loop system consists only of two branches along two parabolas which are asymptotically symmetric relative to the line ${\rm Re}\lambda=-{\rm Im}\lambda$ (the 135$ \circ line in the second quadrant). The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. The Riesz basis property and exponential stability of the system are then proved. Finally we show that the semigroup, generated by the system operator, is of Gevrey class $\delta2$. A numerical computation is presented for the distributions of the spectrum of the closed-loop system.

AB - We study stability of a Schrdinger equation with a collocated boundary feedback compensator in the form of a heat equation with a collocated input/output pair. Remarkably, exponential stability is achieved for both positive and negative gains, namely, as long as the gain is non-zero. We show that the spectrum of the closed-loop system consists only of two branches along two parabolas which are asymptotically symmetric relative to the line ${\rm Re}\lambda=-{\rm Im}\lambda$ (the 135$ \circ line in the second quadrant). The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. The Riesz basis property and exponential stability of the system are then proved. Finally we show that the semigroup, generated by the system operator, is of Gevrey class $\delta2$. A numerical computation is presented for the distributions of the spectrum of the closed-loop system.

KW - Gevrey class

UR - http://www.scopus.com/inward/record.url?scp=84855396713&partnerID=8YFLogxK

U2 - 10.1109/TAC.2011.2164299

DO - 10.1109/TAC.2011.2164299

M3 - Article

AN - SCOPUS:84855396713

SN - 0018-9286

VL - 57

SP - 179

EP - 185

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 1

M1 - 5981378

ER -