This expository talk describes several recent results in the areas of stability and stabilization of nonlinear control systems.
The first part will survey relationships between the existence of feedback stabilizers and control-Lyapunov functions (cLf's). The focus will be on correspondences between, respectively,
Time permitting, the second part of the talk will shift into the effect of "large" disturbances, and notions of stability that take into account external perturbations. The concepts of input-to-state stability (ISS), as well as the more recently introduced integral input-to-state stability (IISS), provide a theoretical framework in which to analyze the issues that arise in this context. The focus will be on dissipation (Lyapunov-like) characterizations of these properties.