This paper deals with boundary feedback control for a fractional reaction-diffusion equation with varying coefficient coupled with fractional ordinary differential equations with delay, which is a generalization of integer order coupled system. By designing a state feedback controller, we transform an unstable system into an asymptotic stable system via the backstepping method. The exact solution of the target system is given by the Prabhakar function. We also obtain the exact solution of the original system with the help of the invertible coordinate transformation. Furthermore, by the fractional Halanay’s inequality, we structure a Lyapunov functional to prove the asymptotic stability of the given system. Finally, a numerical simulation example is provided to illustrate the applications of our results.

本文研究了变系数分数阶反应扩散方程与带分数阶常微分方程的时滞系统的边界反馈控制，它是整数阶耦合系统的推广。通过设计状态反馈控制器，我们通过反步方法将一个不稳定的系统转化为一个渐近的稳定系统。目标系统的确切解决方案由Prabhakar函数给出。我们还借助可逆坐标变换获得了原始系统的精确解。此外，通过分数阶Halanay不等式，我们构造了Lyapunov函数，以证明给定系统的渐近稳定性。最后，提供了一个数值仿真示例来说明我们的结果的应用。