The generalized H-2 optimal control problem for a linear time-invariant system is one in which the conventional H-2 norm is replaced by an operator norm. The closed-loop system is described in terms of a mapping between the space of time-domain input disturbances in L(2) and the space of time-domain regulated outputs in L(infinity). A minimum of this norm is then sought over all stabilizing controllers. It is shown that optimal controllers for such problems have the structure of a Kalman filter with estimated state feedback, where the feedback gains are obtained from the solution to a weighted LQR problem. A computational algorithm is presented to determine the weights in this LQR problem, and examples are given which demonstrate various problems which may arise in obtaining the optimal weights. In particular, it is shown that the generalized H-2 problem may involve the solution to a singular LQR problem.