Feedback law to stabilize linear infinite-dimensional systems

We design a new feedback law to stabilize the linear infinite-dimensional control system, where the state operator generates a \begin{document}$ C_0 $\end{document}-group, and the control operator is unbounded. Our feedback law is based on the integration of a mutated Gramian operator-valued function. In the structure of the aforementioned mutated Gramian operator, we utilize the weak observability inequality in [21,13] and borrow some idea used to construct generalized Gramian operators in [11,23,24]. Unlike most related works where the exact controllability is required, we only assume the above-mentioned weak observability inequality, which is equivalent to the stabilizability of the system.