A boundary element method based on Cauchy integrals for some linear quadratic boundary control problems on a circle

For certain types of elliptic boundary control problems, the boundary element method has considerable advantage over the traditional finite element or finite difference methods because of the reduction of dimensionality in computations. In this paper we examine a variant of such boundary integral methods based on Cauchy integrals. The cost functional here contains only finitely many quadratic terms related to sensory data at those finite interior points. We see that the numerical efficiency of this approach hinges largely on the complexity of the inverse of a certain boundary integral operator. In the case of a circle, such an inverse is readily obtainable and entire computations require only a small effort to yield useful numerical information about the optimal control. Other general situations are also discussed.