In this book we consider the application of boundary integral equation
methods to the solution of boundary control problems governed by
boundary value problems of linear second order elliptic and parabolic
partial differential equations. The use of the related energy spaces
for the Dirichlet control allows to consider a standard variational
formulation. Moreover, it shows the proper mapping properties which link
the Dirichlet and Neumann data in the optimality condition by using
some appropriate operators. In the case of box constraints the
optimality condition is a variational inequality in $H^{1/2}(Gamma)$
which can be written as a Signorini boundary value problem with
bilateral constraints.
Since the unknown function in boundary control
problems is to be found on the boundary of the computational domain,
the use of boundary element methods seems to be a natural choice. In
particular, the solutions of both the state and adjoint boundary value
problems are represented by surface and volume potentials. Since the
state enters the adjoint problem as volume density, we apply integration
by parts to replace these volume potentials by surface potentials. This
results in a system of boundary integral equations. In the case of
parabolic boundary control problems we use an auxiliary function which
relates to the fundamental solution of the heat equation, to get rid of
volume potentials. We derive the unique solvability and study the
boundary element discretizations of the optimality system. Then, we
prove stability and related error estimates. Some numerical examples are
tested to confirm the theoretical results.