This paper shows how H∞ near-optimal finite-dimensional compensators may be designed for linear time invariant (LTI) infinite-dimensional plants subject to convex constraints. The infinite-dimensional plant is approximated by a finite dimensional approximant. The Youla parameterization is used to parameterize the set of all stabilizing LTI controllers and formulate a weighted mixed-sensitivity H∞ optimization that is convex in the Youla Q-Parameter. A finite-dimensional (real-rational) stable basis is used to approximate the Q-parameter. By so doing, we transform the associated infinite-dimensional optimization problem to a finite-dimensional optimization problem involving a search over a finite-dimensional parameter space. In addition to solving weighted mixed-sensitivity H∞ control system design problems, subgradient concepts are used to directly accommodate timedomain specifications (e.g. peak value of control action) in the design process. As such, we provide a systematic design methodology for a large class of infinite-dimensional plant control system design problems. In short, the approach taken permits a designer to address control system design problems for which no direct method exists. Convergence results are presented. Illustrative examples for thermal, structural, and aircraft systems are provided.