In this paper we analyze constrained optimal Dirichlet boundary control problems subject to the linear heat equation. We propose using boundary integral equations to solve the coupled optimality system, and we present results on unique solvability and related a~priori error estimates for a symmetric Galerkin boundary element method. A numerical example confirms the analytical results.
Thanh Xuan Phan. Olaf Steinbach. "Boundary element methods for parabolic boundary control problems." J. Integral Equations Applications 26 (1) 53 - 90, SPRING 2014. https://doi.org/10.1216/JIE-2014-26-1-53