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Following work carried out earlier on linear-quadratic optimal control for linear finite dimensional stationary systems we report,in this article,on extension of some of those results to certain infinite dimensional systems; in particular a class of PDE systems of elliptic type.These systems are studied in the now familiar framework developed by J.L.Lions and E.Magenes,specialized to a subclass of such systems important in a variety of applications.As an extended example this paper studies an optimal redistribution problem in a groundwater flow system goved by Darcy’s equation,presenting both analytic and computational work related to such problems.