We analyze a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the mu-synthesis problem of H-infinity control in which mu is the spectral radius. Given n distinct points gimel(1), ..., gimel(n) in the unit disc and 2 x 2 nonscalar complex matrices W-1, ..., W-n, the problem is to determine whether there is an analytic 2 x 2 matrix function F on the disc such that F(gimel(j)) = W-j for each j and the supremum of the spectral radius of F(gimel) is less than 1 for gimel in the disc. The condition is that the minimum of a quadratic function of pairs of positive 3n-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.