Transformation of optimal control problems with a state constraint avoiding interior boundary conditions

A well-known problem in constrained optimal control is the presence of interior boundary conditions for constrained arcs which require a-priori knowledge of the optimal solution. This paper presents a saturation function method to transform an optimal control problem (OCP) with a state constraint into an unconstrained OCP in new coordinates. The approach allows a tangential entry and exit of constrained arcs without involving interior boundary conditions. An additional regularization term is used in the new OCP to avoid singular arcs which correspond to constrained arcs in the original OCP. Interestingly, the continuity order of the saturation function plays an important role for the existence of bounded trajectories which represent inverse images of the optimal constrained solution in the original coordinates.