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We study a PDE-constrained optimal control problem that involves functions ofbounded variation as controls and includes the TV seminorm of the control inthe objective. We apply a path-following inexact Newton method to the problemsthat arise from smoothing the TV seminorm and adding an $H^1$ regularization.We prove in an infinite-dimensional setting that, first, the solutions of theseauxiliary problems converge to the solution of the original problem and,second, that an inexact Newton method enjoys fast local convergence whenapplied to a reformulation of the optimality system in which the controlappears as implicit function of the adjoint state. We show convergence of aFinite Element approximation, provide a globalized preconditioned inexactNewton method as solver for the discretized auxiliary problems, and embed itinto an inexact path-following scheme. We construct a two-dimensional testproblem with fully explicit solution and present numerical results toillustrate 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 areomitted in the journal's version.

A path-following inexact Newton method for PDE-constrained optimal control in BV (extended version)