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This paper shows how convex optimization may be used to design near-optimal finite-dimensional compensators for stable linear time invariant (LTI) infinite dimensional plants. The infinite dimensional plant is approximated by a finite dimensional transfer function matrix. 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 optimization problem from an infinite dimensional optimization problem involving a search over stable real-rational transfer function matrices in H∞ to a finite-dimensional optimization problem involving a search over a finite-dimensional space. In addition to solving weighted mixed sensitivity H∞ control system design problems, it is shown how subgradient concepts may be 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, illustrative examples are provided.