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This paper presents a study on the optimization of systems with structured uncertainty, whose inputs and outputs can be exhaustively described in the probabilistic sense. By propagating the uncertainty in the space of the probability density functions and the moments, optimization problems that pursue performance, robustness and reliability based designs are studied. Applications to static optimization and stability control are used to illustrate the relevance of incorporating uncertainty in the early stages of the design. Several examples that admit a full probabilistic description of the output in terms of the design variables and the statistics of the uncertain inputs are used to elucidate the features of the generic problem and its solution.