The goal of a boundary control or optimization problem for a fluid flow is to achieve some desired objective by the application of a control mechanism along a portion of the boundary of the flow domain. Among the objectives of interest are some related to drag reduction, some to the avoidance of high temperatures along boundary surfaces and some to conformation to a desired flow pattern. Possible control mechanisms are injected or suction of fluid through boundary orifices and heating or cooling along boundary surfaces. The objective is met through the minimization of an associated cost functional. The method of Lagrange multipliers is used to derive a system of partial differential equations whose solutions provide the optimal states and controls. No simplifications concerning the flow are invoked, i.e. the full Navier-Stokes equations are employed. Finite element algorithms for approximating the optimal states and controls are discussed, as is the accuracy of approximations resulting from these algorithms. Details are provided for the example of using heating and cooling controls in order to avoid hot spots along bounding surfaces. © 1993.