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For a general model of a linear distributed parameter system, the problem of minimizing the norm of the difference between desired system response and obtainable system response is considered. Here the control input is constrained to be bounded in magnitude. An optimal solution is shown to exist, and an optimal solution in a class of controls dense in the constraint set is shown to exist. This latter class is characterized by parameters whose values are obtainable by a convex programming algorithm presented in the paper. The technique developed can also be directly applied to lumped parameter systems, lumped parameter driven distributed parameter systems and the optimum magnitude constrained input final value control problem for any of the preceding.