THE APPROXIMATION OF BOUNDARY CONTROL-PROBLEMS FOR FLUID-FLOWS WITH AN APPLICATION TO CONTROL BY HEATING AND COOLING

The goal of a boundary control or optimization problem for a fluid flo w is to achieve some desired objective by the application of a control mechanism along a portion of the boundary of the flow domain. Among t he 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 a re injection 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 equatio ns whose solutions provide the optimal states and controls. No simplif ications concerning the flow are invoked, i.e. the full Navier-Stokes equations are employed. Finite element algorithms for approximating th e optimal states and controls are discussed, as is the accuracy of app roximations resulting from these algorithms. Details are provided for the example of using heating and cooling controls in order to avoid ho t spots along bounding surfaces.