Invariant manifolds and 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 stabilizing values-the 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 arty time without destroying the closed-loop system stability.