The problem of minimizing the L∞-norm of some stable rational matrix E(s) subject to two basic types of matrix interpolation constraints is considered: given anti-stable rational matrices N¯(s) and M¯(s), the interpolating matrix £(s) is required to satisfy either the condition N¯(s)E(s) = M¯(s) or the condition that N¯(s)E(s) has an unstable projection equal to M¯(s). These two kinds of constraints are directly related to model-matching problems arising from H∞optimal control. Specific conditions on N¯(s) and M¯(s) for the interpolation problems to be well-posed are discussed and closed-form characterizations for both optimal and suboptimal solutions in E(s) are provided. The analysis is based on a state-space setting and the results are suitable for computational purposes.