This paper concerns the problem of boundary time-varying feedback controller for fixed-time stabilization of a linear parabolic distributed parameter system with spatially and temporally varying reactivity. By utilizing the continuous backstepping approach, the invertible Volterra integral transformation with the time-dependent gain kernel is introduced to convert the closed-loop system into a target system with a time-dependent coefficient. Meanwhile, the convergence of the target system within the prescribed time is guaranteed via the Lyapunov method. The well-posedness of the resulting kernel partial differential equations is also proven by exploiting the method of successive approximation. In addition, the growth-in-time of the kernel functions is estimated by applying the generalized Laguerre polynomials and the modified Bessel functions. Subsequently, the fixed-time stability of the closed-loop system under state feedback control within the prescribed time is proven by using the fixed-time stability of the target system and the time-varying kernel functions. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control method.

本文涉及边界时变反馈控制器问题，用于对具有空间和时间变化反应性的线性抛物线分布参数系统进行固定时间稳定。利用连续反步法，引入具有时间相关增益内核的可逆沃尔泰拉积分变换，将闭环系统转换为具有时间相关系数的目标系统。同时，通过李雅普诺夫方法保证目标系统在规定时间内的收敛。所得到的核偏微分方程的适定性也通过利用逐次逼近的方法得到证明。此外，通过应用广义拉盖尔多项式和修正的贝塞尔函数来估计核函数的随时间增长。随后，利用目标系统的固定时间稳定性和时变核函数证明了闭环系统在状态反馈控制下在规定时间内的固定时间稳定性。最后，提供了一个数值例子来说明所提出的控制方法的有效性。