  It is well known from the seminal Brockett's theorem that the openness
property of the mapping on the right-hand side of a given nonlinear ODE control
system is a necessary condition for the existence of locally asymptotically
stabilizing continuous stationary feedback laws. However, this condition fails
to be sufficient for such a feedback stabilization. In this paper we develop an
approach of variational analysis to continuous feedback stabilization of
nonlinear control systems with replacing openness by the linear openness
property, which has been well understood and characterized in variational
theory. It allows us, in particular, to obtain efficient conditions via the
system data supporting the sufficiency in Brockett's theorem and ensuring local
exponential stabilization by means of continuous stationary feedback laws.
Furthermore, we derive new necessary conditions for local exponential and
asymptotic stabilization of continuous-time control systems by using both
continuous and smooth stationary feedback laws and establish also some
counterparts of the obtained sufficient conditions for local asymptotic
stabilization by continuous stationary feedback laws in the case of nonlinear
discrete-time control systems.
