Boundary stabilization of non-diagonal systems by proportional feedback forms

In this work, we are concerned with the problem of boundary exponential stabilization, in a Hilbert space \begin{document}$ H $\end{document} , of parabolic type equations, namely equations for which their linear parts generate analytic \begin{document}$ C_0- $\end{document} semigroups. We consider the case where the projection of the linear leading operator, on a given Riesz basis of \begin{document}$ H $\end{document} , is non-diagonal. We do not assume that the linear operator has compact resolvent. Therefore, the Riesz basis is not necessarily an eigenbasis. The boundary stabilizer is given in a simple linear feedback form, of finite-dimensional structure, involving only the Riesz basis. To illustrate the results, at the end of the paper, we provide an example of stabilization of a fourth-order evolution equation on the half-axis.