  We consider optimal control problems with integer-valued controls and a total
variation regularization penalty in the objective on domains of dimension two
or higher. The penalty yields that the feasible set is sequentially closed in
the weak-$^*$ and closed in the strict topology in the space of functions of
bounded variation.
  In turn, we derive first-order optimality conditions of the optimal control
problem as well as trust-region subproblems with partially linearized model
functions using local variations of the level sets of the feasible control
functions. We also prove that a recently proposed function space trust-region
algorithm -- sequential linear integer programming -- produces sequences of
iterates whose limits are first-order optimal points.
