Topological aspects of universal adaptive stabilization

We consider two related problems in adaptive stabilization without identification. The first problem concerns the sensitivity of closed-loop solutions to small perturbations in initial data. The second problem is to characterize those initial conditions for which the adaptive control algorithm converges to a fixed limit controller which is stabilizing. We prove that for all initial data in some open, dense and full Lebesgue measure set the closed-loop solutions are insensitive to perturbations and the limit controller is stabilizing. Our results are restricted to a class of control algorithms which are typified by the `dense searching controller´ of Martensson (1986) in which the closed-loop gain evolution is piecewise constant