Discontinuous Feedback and Nonlinear Systems *

This tutorial paper is devoted to the controllability and stability of control systems that are nonlinear, and for which, for whatever reason, linearization fails. We begin by motivating the need for two seemingly exotic tools: nonsmooth control-Lyapunov functions, and discontinuous feedbacks. Then, after a (very) short course on nonsmooth analysis, we build a theory around these tools. We proceed to apply it in various contexts, focusing principally on the design of discontinuous stabilizing feedbacks.