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This paper addresses designing finite dimensional linear time invariant (LTI) controllers for infinite dimensional LTI plants subject to HÂ¿ mixed-sensitivity performance objectives and convex constraints. Specifically, we focus on designing control systems for two classes of systems which are generally described by hyperbolic partial differential equations: (1) irrigation systems and (2) hypersonic vehicles with flexible dynamics. The distributed parameter plant is first approximated by a finite dimensional approximant. The Youla parameterization is then used to parameterize the set of all stabilizing LTI controllers and a weighted mixed-sensitivity HÂ¿ optimization is formulated. After transforming the infinite dimensional problem to a finite-dimensional optimization problem, convex is optimization is used to obtain the solution. Subgradient concepts are used to directly accommodate time domain specifications. Illustrative examples for irrigation systems and hypersonic vehicles are provided.