  Local asymptotic stabilizability is a topic of great theoretical interest and
practical importance. Broadly, if a system $\dot{x} = f(x,u)$ is locally
asymptotically stabilizable, we are guaranteed a feedback controller $u(x)$
that forces convergence to an equilibrium for trajectories initialized
sufficiently close to it. A necessary condition was given by Brockett: such
controllers exist only when $f$ is open at the equilibrium. Recently, Gupta,
Jafari, Kipka and Mordukhovich considered a modification to this condition,
replacing Brockett's topological openness by the linear openness property of
modern variational analysis. In this paper, we show that under the linear
openness assumption there is a local diffeomorphism of neighborhoods of the
equilibirum on which the system is exponentially stabilizable by means of
continuous stationary feedback laws. Introducing a transversality property and
relating it to the above diffeomorphism, we prove that linear openness and
transversality on a punctured neighborhood of an equilibrium is sufficient for
local exponential stabilizability of systems with a rank deficient
linearization.The main result goes beyond the usual Kalman and Hautus criteria
for the existence of exponential stabilizing feedback laws, since it allows us
to handle systems for which exponential stabilization is achieved through
higher-order terms. However, it is implemented so far under a rather
restrictive row-rank conditions. This suggests a twofold approach to the use of
these properties: a point-wise version is enough to ensure stability via the
linearization, while a local version is enough to overcome deficiencies in the
linearization.
