Pareto suboptimal controllers versus coalitions of disturbances

We consider a multi-criteria problem of suppressing disturbances with linear feedback with respect to the state or output measured with noise. We assume that the system has $N$ potentially possible inputs for disturbances from given classes, and the criteria are induced norms of operators generated by the system from the corresponding input to the common target output. We obtain necessary Pareto optimality conditions. We show that based on scalar optimization of the suppression level for the disturbances that act on all inputs we can synthesize Pareto suboptimal controllers whose relative losses compared to Pareto optimal controllers do not exceed $1-\sqrt N/N$. Our results generalize to the case when disturbances from different classes may form coalitions.