Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrödinger equation subject to boundary control matched disturbance.

*(English)*Zbl 1302.93060Summary: We are concerned with the boundary stabilization of a one-dimensional anti-stable Schrödinger equation subject to boundary control matched disturbance. We apply both the sliding mode control (SMC) and the active disturbance rejection control (ADRC) to deal with the disturbance. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system is proved and the ’reaching condition’ is obtained. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we develop the ADRC to attenuate the disturbance for which the derivative is also supposed to be bounded. Compared with the SMC, the advantage of the ADRC is not only using the continuous control but also giving an online estimation of the disturbance. It is shown that the resulting closed-loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero.

##### MSC:

93B12 | Variable structure systems |

93B35 | Sensitivity (robustness) |

35Q41 | Time-dependent Schrödinger equations and Dirac equations |

93C73 | Perturbations in control/observation systems |

##### Keywords:

Schrödinger equation; sliding mode control; active disturbance rejection control; stability, boundary control; disturbance rejection
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\textit{B.-Z. Guo} and \textit{J.-J. Liu}, Int. J. Robust Nonlinear Control 24, No. 16, 2194--2212 (2014; Zbl 1302.93060)

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