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 observed phenomenon is general, and will arise in many different optimization problems.