  We study a PDE-constrained optimal control problem that involves functions of
bounded variation as controls and includes the TV seminorm of the control in
the objective. We apply a path-following inexact Newton method to the problems
that arise from smoothing the TV seminorm and adding an $H^1$ regularization.
We prove in an infinite-dimensional setting that, first, the solutions of these
auxiliary problems converge to the solution of the original problem and,
second, that an inexact Newton method enjoys fast local convergence when
applied to a reformulation of the optimality system in which the control
appears as implicit function of the adjoint state. We show convergence of a
Finite Element approximation, provide a globalized preconditioned inexact
Newton method as solver for the discretized auxiliary problems, and embed it
into an inexact path-following scheme. We construct a two-dimensional test
problem with fully explicit solution and present numerical results to
illustrate the accuracy and robustness of the approach.
  This is an extended version of the corresponding journal article appearing in
"Computational Optimization and Applications". It contains some proofs that are
omitted in the journal's version.
