Exact solutions to some minimum-time problems and their behavior near inequality state constraints

A heuristic method for generating exact solutions to certain minimum-time problems with inequality state constraints is used to generate solutions to a class of path-planning problems. It is observed that, when the state constraint function has a continuous second derivative, the constraint does not become active for any continuous-time period. Instead, the solution bumps up against the constraint repeatedly at isolated points. The solution method offers some insight into this behavior. It is shown that such a state constraint can become active for a continuous-time period only if the solution path satisfies an overdetermined system of equations. It is argued that the phenomenon is general and will arise in many different optimization problems