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Uniform stabilization of 1-D Schrödinger equation with internal difference-type control

  • * Corresponding author: Xiaorui Wang

    * Corresponding author: Xiaorui Wang 

This project is partially supported by the National Natural Science Foundation in China (NSFC 61773277), and partially supported by NSF of Qinghai Province (2017-ZJ-908)

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  • In this paper, we consider the stabilization problem of 1-D Schrödinger equation with internal difference-type control. Different from the other existing approaches of controller design, we introduce a new approach of controller design so called the parameterization controller. At first, we rewrite the system with internal difference-type control as a cascaded system of a transport equation and Schödinger equation; Further, to stabilize the system under consideration, we construct a target system that has exponential stability. By selecting the solution of nonlocal and singular initial value problem as parameter function and defining a bounded linear transformation, we show that the transformation maps the closed-loop system to the target system; Finally, we prove that the transformation is bounded inverse. Hence the closed-loop system is equivalent to the target system.

    Mathematics Subject Classification: Primary: 35J10, 93C20; Secondary: 93D15.

    Citation:

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  • Figure 1.  The dynamic behaviour of system (1) for $ \alpha = \beta = 0 $

    Figure 2.  The dynamic behaviour of system (1) for $ \alpha = 1, \beta = 0 $ under $ U(t) = -kw(x,t) $

    Figure 3.  The dynamic behaviour of system (1) for $ \alpha = 2, \beta = 1 $ under $ U(t) = -kw(x,t) $

    Figure 4.  The dynamic behaviour of system (1) for $ \alpha = 1, \beta = 0 $ under the control (3)

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