Stabilization of control systems associated with a strongly continuous group

Abstract

This paper is devoted to the stabilization of a linear control system $y^{\prime }=Ay+Bu$ and its suitable non-linear variants where $(A, \cD(A))$ is an infinitesimal generator of a strongly continuous {\it group} in a Hilbert space $\mH$, and $B$ defined in a Hilbert space $\mU$ is an admissible control operator with respect to the semigroup generated by $A$. Let $λ\in \mR$ and assume that, for some {\it positive} symmetric, invertible $Q = Q(λ) \in \cL(\mH)$, for some {\it non-negative}, symmetric $R = R(λ) \in \cL(\mH)$, and for some {\it non-negative}, symmetric $W = W(λ) \in \cL(\mU)$, 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.