Stabilization of control systems associated with a strongly continuous group

This paper is devoted to the stabilization of a linear control system $y^{\mathrm{\prime }}=Ay+Bu$ and its suitable non-linear variants where $(A,\text{cD}(A))$ is an infinitesimal generator of a strongly continuous {\it group} in a Hilbert space $\text{mH}$, and $B$ defined in a Hilbert space $\text{mU}$ is an admissible control operator with respect to the semigroup generated by $A$. Let $\lambda \in \text{mR}$ and assume that, for some {\it positive} symmetric, invertible $Q=Q(\lambda )\in \text{cL}(\text{mH})$, for some {\it non-negative}, symmetric $R=R(\lambda )\in \text{cL}(\text{mH})$, and for some {\it non-negative}, symmetric $W=W(\lambda )\in \text{cL}(\text{mU})$, it holds $$AQ+Q{A}^{\ast }-BW{B}^{\ast }+QRQ+2\lambda Q=0.$$ We then present a new approach to study the stabilization of such a system and its suitable nonlinear variants. Both the stabilization using dynamic feedback controls and the stabilization using static feedback controls in a weak sense are investigated. To our knowledge, the nonlinear case is out of reach previously when $B$ is unbounded for both types of stabilization.