Stabilization and Gevrey Regularity of a Schrödinger Equation in Boundary Feedback With a Heat Equation
Stabilization and Gevrey Regularity of a Schrödinger Equation in Boundary Feedback With a Heat Equation
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DOI：
10.1109/tac.2011.2164299
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发表时间：
2012
影响因子：
6.8
通讯作者：
Jun‐min Wang;B. Ren;M. Krstić
中科院分区：
文献类型：
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作者：
Jun‐min Wang;B. Ren;M. Krstić
我们研究具有并置边界反馈补偿器的薛定谔方程的稳定性，其形式为具有并置输入/输出对的热方程。值得注意的是，只要增益非零，正增益和负增益都可以实现指数稳定性。我们证明闭环系统的频谱仅由沿着两条抛物线的两个分支组成，这两条抛物线相对于线 Reλ = -Imλ（第二象限中的 135° 线）渐近对称。得到了特征值和特征函数的渐近表达式。然后证明了系统的Riesz基性质和指数稳定性。最后我们证明由系统算子生成的半群属于 Gevrey 类 δ >； 2.对闭环系统的频谱分布进行了数值计算。
We study stability of a Schrödinger 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 Reλ = -Imλ (the 135° 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 δ >; 2. A numerical computation is presented for the distributions of the spectrum of the closed-loop system.