We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP) in two ways. We improve a theorem that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike in previous publications the new short and self-contained proof avoids the usage of the Krein-Milman theorem, which is undesirable as it only states the existence of a solution that may switch infinitely often. We prove that this bound depends on the control discretization grid. The rounded solution will be arbitrarily close to the relaxed one, if only the underlying grid is chosen fine enough. A numerical benchmark example illustrates the procedure.
Mathematical Programming A, http://dx.doi.org/10.1007/s10107-010-0405-3