  For a map that is strictly but not strongly convex, model-based gradient
extremum seeking has an eigenvalue of zero at the extremum, i.e., it fails at
exponential convergence. Interestingly, perturbation-based model-free extremum
seeking has a negative Jacobian, in the average, meaning that its (practical)
convergence is exponential, even though the map's Hessian is zero at the
extremum. While these observations for the gradient algorithm are not trivial,
we focus in this paper on an even more nontrivial study of the same phenomenon
for Newton-based extremum seeking control (NESC).
  NESC is a second-order method which corrects for the unknown Hessian of the
unknown map, not only in order to speed up parameter convergence, but also (1)
to make the convergence rate user-assignable in spite of the unknown Hessian,
and (2) to equalize the convergence rates in different directions for
multivariable maps. Previous NESC work established stability only for maps
whose Hessians are strictly positive definite everywhere, so the Hessian is
invertible everywhere. For a scalar map, we establish the rather unexpected
property that, even when the map behind is strictly convex but not strongly
convex, i.e., when the Hessian may be zero, NESC guarantees practical
asymptotic stability, semiglobally. While a model-based Newton-based algorithm
would run into non-invertibility of the Hessian, the perturbation-based NESC,
surprisingly, avoids this challenge by leveraging the fact that the average of
the perturbation-based Hessian estimate is always positive, even though the
actual Hessian may be zero.
