An Example of a Globally Stabilizing

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