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 closedloop 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.
The state observation problem is tackled for a system ofncoupled reaction-diffusion PDEs, possessing the same diffusivity parameter and equipped with boundary sensing devices. Particularly, a backstepping-based observer is designed and the exponential stability of the error system is proven with an arbitrarily fast convergence rate. The transformation kernel matrix is derived in the explicit form by using the method of successive approximations, thereby yielding the observer gains in the explicit form, too. Simulation results support the effectiveness of the suggested design.
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