Abstract

In this paper, we investigate the stabilization problem of a cascade of a fractional ordinary differential equation (FODE) and a fractional reaction–diffusion (FRD) equation where the interconnections are of Neumann type. We exploit the partial differential equation backstepping method for designing a controller, which guarantees the Mittag–Leffler stability of the FODE-FRD cascade. Moreover, we propose an observer that is Mittag–Leffler convergent. Also, we propose an output feedback boundary controller, and we prove that the closed-loop FODE-FRD system is Mittag–Leffler stable in the sense of the corresponding norm. Finally, numerical simulations are presented to verify the results.

中文翻译：

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分数阶普通方程与反应扩散方程耦合的基于观测器的输出反馈控制设计

摘要

在本文中，我们研究了分数阶常微分方程（FODE）和分数反应扩散方程（FRD）的级联的镇定问题，其中的互连是Neumann型的。我们利用偏微分方程反推方法设计控制器，从而保证了FODE-FRD级联的Mittag-Leffler稳定性。此外，我们提出了一个Mittag-Leffler收敛的观测器。此外，我们提出了一种输出反馈边界控制器，并且证明了在相应范数的意义上，闭环FODE-FRD系统是Mittag–Leffler稳定的。最后，通过数值模拟验证了结果。