Asymptotic properties of adaptive nonlinear stabilizers

A classical question in adaptive control is that of convergence of the parameter estimates to constant values in the absence of persistent excitation. We provide an affirmative answer for a class of adaptive stabilizers for nonlinear systems. Then we study their asymptotic behavior by considering the problem of whether the parameter estimates converge to values which would guarantee stabilization if used in a nonadaptive controller. We approach this problem by studying invariant manifolds and show that, except for a set of initial conditions of Lebesgue measure zero, the parameter estimates do converge to stabilizing values. Finally, we determine a (sufficiently large) time instant after which the adaptation can be disconnected at any time without destroying the closed-loop system stability.