In this note we investigate the following questions: given a (finite-dimensional) linear time-invariant (LTI) multivariable system and a constant desired value for its output, say y(*). Assume there is no assignable equilibrium point corresponding to y(*). How "close" to y(*) can we ultimately keep the output using LTI static state-feedback stabilizing controllers? Can this neighborhood of y(*) be reduced with dynamic, nonlinear, time-varying controllers? Our main contributions are the proof that the optimal ultimate boundedness neighborhood is achieved with LTI static state-feedback, the explicit computation of the neighborhood's size and the proof, under some reasonable rank assumptions, that the system has non-assignable values for the output if and only if it has a transmission zero at zero. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability. (c) 2007 Elsevier Ltd. All rights reserved.