In this paper, we consider control problems where the objective is to minimize the worst-case variance of the regulated output over the class of all exogenous signals that meet partially specified statistical information. State-space results for analysis and state feedback design are presented using finite-dimensional convex optimization problems of the max-min type where the inner minimization problem involves linear matrix inequalities (LMI). These results can be viewed from a theoretical perspective as a bridge between the H-2 and H-infinity design methodologies. For clarity of exposition and a more transparent signal theoretic interpretation, only the single-input-single-output case is treated in this paper.