On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters

We consider the problem of boundary stabilization for a system of n coupled parabolic linear PDEs. Particularly, we design a state feedback law with actuation on only one end of the domain and prove exponential stability of the closed-loop system with an arbitrarily fast convergence rate. The backstepping method is used for controller design, and the transformation kernel matrix is derived in explicit form of series of matrix Bessel functions by using the method of successive approximations to solve the corresponding PDE. Thus, the proposed control law becomes available in explicit form. Simulation results support the effectiveness of the suggested design.