This paper focuses on ℋ∞ near-optimal finite-dimensional compensator design for linear time invariant (LTI) distributed parameter plants subject to convex constraints. The distributed parameter plant is first approximated by a finite dimensional approximant. For unstable plants, the coprime factors are approximated by their finite dimensional approximants. The Youla parameterization is then 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, the associated infinite-dimensional optimization problem is transformed 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, overshoot) in the design process. As such, a systematic design methodology is provided for a large class of distributed parameter plant control system design problems. Convergence results are presented. An illustrative examples for a hypersonic vehicle is provided. In short, the approach taken permits a designer to address control system design problems for which no direct method exists.