The stabilization problem for the Schrödinger equation with an input time delay is considered from the view of system equivalence. First, a linear transform from the original system into an exponentially stable system with arbitrary decay rate, also called “target system”, is introduced. The linear transform is constructed via a kind of Volterra-type integration with singular kernels functions. As a result, a feedback control law for the original system is obtained. Secondly, a linear transform from the target system into the original closed-loop system is derived. Finally, the exponential stability with arbitrary decay rate of the closed-loop system is obtained through the established equivalence between the original closed-loop system and the target one. The authors conclude this work with some numerical simulations giving support to the results obtained in this paper.

从系统等价的角度考虑了带有输入时滞的Schrödinger方程的镇定问题。首先，介绍了从原始系统到具有任意衰减率的指数稳定系统的线性转换，也称为“目标系统”。线性变换是通过一种具有单一核函数的Volterra型集成来构造的。结果，获得了原始系统的反馈控制律。其次，推导了从目标系统到原始闭环系统的线性变换。最后，通过建立原始闭环系统与目标闭环系统之间的等价关系，获得了闭环系统具有任意衰减率的指数稳定性。