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 and assume that, for some positive symmetric, invertible Q = Q () (), for some non-negative, symmetric R = R () (), and for some non-negative, symmetric W = W () (), it holds A Q + Q A^* - B W B^* + Q R Q + 2 Q = 0 in the sense that Qx, A^*y _ + A^*x, Q y _ - W B^*x, B^*y _ + R Qx, Q y _ + 2 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.