Stabilization of control systems associated with a strongly continuous group

This paper is devoted to the stabilization of a linear control system $y' = A y + B u$ and its suitable non-linear variants where (A, (A)) is an infinitesimal generator of a strongly continuous { group} in a Hilbert space , and B defined in a Hilbert space is an admissible control operator with respect to the semigroup generated by A. Let λ ∈ and assume that, for some { positive} symmetric, invertible Q = Q(λ) ∈ (), for some { non-negative}, symmetric R = R(λ) ∈ (), and for some { non-negative}, symmetric W = W(λ) ∈ (), it holds A Q + Q A^* - B W B^* + Q R Q + 2 λ Q = 0 in the sense that $$ Qx, A^*y _{} + A^*x, Q y _{} - W B^*x, B^*y _{} + R Qx, Q y _{} + 2 λ Q x, y _{}= 0 ∀ \, x, y ∈ (A^*), $$ where $A^*$ is the adjoint of $A$ and $(A^*)$ is its domain. We present a new method 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 stabilization by dynamic feedback controls is new even in the linear setting. The nonlinear case is out of reach previously when B is unbounded for both types of stabilization. Consequently, we derive that if the control system is exactly controllable in some positive time, then it is rapidly stabilizable.