The present paper addresses the problem of the existence of an ( output) feedback law that asymptotically steers to zero prescribed outputs, while keeping all state variables bounded, for any initial conditions in a given compact set. The problem can be viewed as an extension of the classical problem of semiglobally stabilizing the trajectories of a controlled system to a compact set. The problem also encompasses a version of the classical problem of output regulation. Under only a weak minimum phase assumption, it is shown that there exists a controller solving the problem at hand. The paper is deliberately focused on theoretical results regarding the existence of such a controller. Practical aspects involving the design and the implementation of the controller are left to a forthcoming work.