An optimality system of equations for the optimal control problem governed by Helmholtz-type equations is derived. By the associated first-order necessary optimality condition, we obtain the conjugate gradient method (CGM) in the continuous case. Introducing the sequence of higher-order fundamental solutions, we propose an iterative algorithm based on the conjugate gradient-boundary element method using the multiple reciprocity method (CGM+MRBEM) for solving the discrete control input. This algorithm has an advantage over that of the existing literatures because the main attribute (the reduced dimensionality) of the boundary element method is fully utilized. Finally, the local error estimates for this scheme are obtained, and a test problem is given to illustrate the efficiency of the proposed method. © 2007 Elsevier Inc. All rights reserved.
Li, B., & Liu, S. (2007). Conjugate gradient-boundary element solution for distributed elliptic optimal control problems. Journal of Mathematical Analysis and Applications, 335(2), 1219–1237. https://doi.org/10.1016/j.jmaa.2007.02.003