Explicit control law of a coupled reaction-diffusion process

Stabilization of a coupled linear plant and reaction-diffusion process by boundary control was considered in a paper by Zhao and Xie. In this paper, backstepping transformation with a kernel function and a vector-valued function was introduced to design a control law. For the situation without heat resource in the reaction-diffusion process, the kernel function and the vector-valued function in the transformation had been obtained, and the explicit control law had been established. However, for the reaction-diffusion process with heat resource, the explicit solutions of the kernel function and the vector-valued function in the transformation have not been obtained. Thus, an explicit control law has not been established. In the present paper, the explicit solutions of the kernel function and the vector-valued function in the transformation are obtained through some mathematical skills and complicated computation, and the explicit control law has been established.