Feedback law to stabilize linear infinite-dimensional systems

Abstract

<p style='text-indent:20px;'>We design a new feedback law to stabilize the linear infinite-dimensional control system, where the state operator generates a <inline-formula><tex-math id="M1">\begin{document}$ C_0 $\end{document}</tex-math></inline-formula>-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 [<xref ref-type="bibr" rid="b21">21</xref>,<xref ref-type="bibr" rid="b13">13</xref>] and borrow some idea used to construct generalized Gramian operators in [<xref ref-type="bibr" rid="b11">11</xref>,<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b24">24</xref>]. 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.</p>