An example of a globally stabilizing adaptive controller with a generically destabilizing parameter estimate

In this note, we consider the question of whether an adaptive controller can converge to a nonadaptive stabilizing controller. Specifically, we show, for a class of back-stepping controllers with adaptive tuning functions, that the set of initial conditions in a state and estimation parameter for which the estimation parameter converges to a parameter which produces a destabilizing controller can have a nonempty interior and, consequently, a nonzero Lebesgue measure. This surprising result is proved by way of a simple example with a quadratic nonlinearity