This paper shows how ℋ∞ near-optimal finite-dimensional compensators may be designed for stable 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 from 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 time-domain 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. An illustrative example for a thermal diffusion process is also provided.