  The present paper addresses the problem of existence of an (output) feedback
law to the purposes of asymptotically steering to zero a given controlled
variable, 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 semi-globally stabilizing the trajectories of a controlled
system to a compact set. The problem also encompasses a version of the
classical problem of output regulation. Assuming only the existence of a
feedback law that keeps the trajectories of the zero dynamics of the controlled
plant bounded, 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 controller. Practical aspects involving the design and the
implementation of the controller are left to a forthcoming work.
