Variations of the standard H infinity and H2 control problems are presented. State-space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while (i) satisfying an H infinity-Frobenius norm bound or (ii) minimizing the H2 norm of the closed loop. The optimal H infinity-Frobenius case is analysed and it is shown that the optimal controller is often unique, in contrast to the standard H infinity control problem. The solutions of an algebraic Riccati equation are examined in the H2 case to reveal the structure of the H2 optimal controller.