  We propose a trust-region method that solves a sequence of linear integer
programs to tackle integer optimal control problems regularized with a total
variation penalty.
  The total variation penalty allows us to prove the existence of minimizers of
the integer optimal control problem. We introduce a local optimality concept
for the problem, which arises from the infinite-dimensional perspective. In the
case of a one-dimensional domain of the control function, we prove convergence
of the iterates produced by our algorithm to points that satisfy first-order
stationarity conditions for local optimality. We demonstrate the theoretical
findings on a computational example.
