We present in this paper preliminary results and research directions concerning the output regulation problem for over-actuated linear systems. The focus of the paper is on the characterization of the solution of the full-information regulator problem for systems which are right-invertible but not left-invertible, where the input operator is injective. The intrinsic redundancy in the plant model is exploited by parameterizing all solutions of the ensuing regulator equations and performing a static or dynamic optimization on the space of solutions. This approach effectively shapes the non-unique steady-state of the system so that the long-term behavior optimizes a given performance index. In particular, nonlinear cost functions that account for constraints on the inputs are considered. Examples are given to illustrate and validate the proposed methodology.