Noncollocated exponential stabilization for a class of parabolic system with heterogeneous material and boundary recycle

In this article, we consider the noncollocated exponential stabilization of a class of parabolic system with heterogeneous material and boundary recycle, where every parabolic equation has different diffusion coefficient, the information exchange occurs at the boundaries of adjacent two equations, and also exists between the first and last equations. The control is applied at the left boundary x=0 and the temperature of the right boundary x=Nπ is set as the measurement point of the whole system. We design the noncollocated static feedback control and make a detailed spectral analysis to investigate the eigenvalues distribution of the closed-loop system. Then we prove the Riesz basis property of the corresponding eigenfunctions and the system operator is a spectral operator. Finally, we show the system is well-posed and exponentially stable in the state space by the spectrum-determined growth condition.