In this paper we consider a prototypical model matching problem where the various mappings involved are systems that switch arbitrarily among n given linear time invariant (LTI) systems. The interest is placed on minimizing the worst case ℓ∞ -induced norm of this model matching system over all possible switchings. This minimization is performed over all Youla-Kucera parameters Q that switch causally in time over n (stable) LTI systems. For the particular set-up at hand, it is shown that the optimal Youla-Kucera parameter Q need not depend on the switching sequence in the case of partially matched switching, and that it can be obtained as an LTI solution to an associated standard ℓ1 optimization. In the case of matched switching, two convergent sequences to the optimal solution from above and below respectively are formulated in linear programming, and an approximate solution with any given precision is possible by finite truncation.