On nonexistence of boundary arcs in control problems with bounded state variables

When solving optimal control problems with bounded state variables, one must determine whether the optimal trajectory intersects the boundary only at isolated points in time (boundary point) or remains on the boundary for a nonzero length of time (boundary arc). Previously, this determination has been made by trial and error. The task is complicated by the fact that the necessary conditions in common use for these problems assume that the solution has a boundary arc, and can thus yield a boundary arc when the solution has no boundary arc. In this paper the necessary conditions of [1] are used to derive conditions under which the optimal trajectory cannot have a boundary arc. These conditions include the condition for no boundary arcs developed in [1] as a special case. The application of these conditions is illustrated via several examples.