FEEDBACK STABILIZATION OF NONLINEAR SYSTEMS Eduardo D. Sontag Abstract This paper surveys some well-known facts as well as some recent developments on the topic of stabilization of nonlinear systems. 1 Introduction In this paper we consider problems of local and global stabilization of control systems ˙ x = f (x, u) , f (0, 0) = 0 (1) whose states x(t) evolve on IR n and with controls taking values on IR m , for some integers n and m. The interest is in finding feedback laws u = k(x) , k(0) = 0 which make the closed-loop system ˙ x = F (x)= f (x, k(x)) (2) asymptotically stable about x = 0. Associated problems, such as those dealing with the response to possible input perturbations u = k(x)+ v of the feedback law, will be touched upon briefly. We assume that f is smooth (infinitely differentiable) on (x, u), though much less, –for instance a Lipschitz condition,– is needed for many results. The discussion will emphasize intuitive aspects, but we shall state the main results as clearly as possible. The references cited should be consulted, however, for all technical details. Some comments on the contents of this paper: • We do not consider control objectives different from stabilization, such as decoupling or disturbance rejection. • Except for some remarks, we consider only state (rather than output) feedback. • The survey talk centers on questions of possible regularity (continuity, smoothness) of k. This focus leads to natural mathematical questions, and it may be argued that that regular feedback is more “robust” in various senses. But –and to some extent this is emphasized by those negative results that are presented– it is often the case that discontinuous control laws must be considered (sliding mode controllers, or piecewise smooth feedback, for instance). In addition, non-continuous-time feedback (sampled control, pulse-width modulation), is often used in practice and is also not covered here.