Asymptotic controllability and robust asymptotic stabilizability

This paper deals with asymptotically controllable systems for which there exists no smooth stabilizing state feedback. To investigate the robustness asymptotic stabilization property, a new class of hybrid feedbacks (with a continuous component and a discrete one) is introduced: the hybrid patchy feedbacks. The notion of solutions is a generalization of pi-solutions and Euler solutions. It is proved that the origin of all globally asymptotically controllable systems can be globally asymptotically stabilized via a hybrid feedback with robustness with respect to measurement noise, actuator errors, and external disturbances.