Further Constructions of Control-Lyapunov Functions and Stabilizing   Feedbacks for Systems Satisfying the Jurdjevic-Quinn Conditions
  For a broad class of nonlinear systems, we construct smooth control-Lyapunov functions whose derivatives along the trajectories of the systems can be made negative definite by smooth control laws that are arbitrarily small in norm. We assume our systems satisfy appropriate generalizations of the Jurdjevic-Quinn conditions. We also design state feedbacks of arbitrarily small norm that render our systems integral-input-to-state stable to actuator errors. 
Frederic Mazenc and Michael Malisoff