Lyapunov Functions and Feedback in Nonlinear Control

The method of Lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. For nonlinear systems, it turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth. We synthesize in this article a number of recent developments bearing upon the regularity properties of Lyapunov functions. A novel feature of our approach is that the guidability and stability issues are decoupled. For each of these issues, we identify various regularity classes of Lyapunov functions and the system properties to which they correspond. We show how such regularity properties are relevant to the construction of stabilizing feedbacks. Such feedbacks, which must be discontinuous in general, are implemented in the sample-and-hold sense. We discuss the equivalence between open-loop controllability, feedback stabilizability, and the existence of Lyapunov functions with appropriate regularity properties. The extent of the equivalence confirms the cogency of the new approach summarized here.