Boundary feedback stabilisation for the time fractional‐order anomalous diffusion system

Abstract

In this study, the authors attempt to explore the boundary feedback stabilisation for an unstable heat process described by fractional‐order partial differential equation (PDE), where the first‐order time derivative of normal reaction–diffusion equation is replaced by a Caputo time fractional derivative of order α∈(0, 1]. By designing an invertible coordinate transformation, the system under consideration is converted into a Mittag–Leffler stability linear system and the boundary stabilisation problem is transformed into a problem of solving a linear hyperbolic PDE. It is worth mentioning that with the help of this invertible coordinate transformation, they can explicitly obtain the closed‐loop solutions of the original problem. The output feedback problem with both anti‐collocated and collocated actuator/sensor pairs in one‐dimensional domain is also presented. A numerical example is given to test the effectiveness of the authors' results.