In this paper we consider the problem of minimizing the H ∞ -norm of the sum of two systems which are interconnected with two different controller components. We sketch two specific applications of this decentralized control problem. As the main contribution, we reveal that this structured controller design problem admits a full LMI solution if only one of the systems in the interconnection obeys a one-block hypothesis. As an important structural insight, our solution allows to give explicit bounds on the McMillan degrees of both controller components in terms of the original system description if approaching optimality. This might be an important step towards overcoming the one-block hypothesis which could lead to direct LMI solutions of general multi-objective H ∞ -control problems.