Chattering Phenomena in Time-Optimal Control for High-Order Chain-of-Integrators Systems with Full State Constraints

Time-optimal control for high-order chain-of-integrators systems with full state constraints and arbitrary given terminal states remains an open and challenging problem in optimal control theory domain. However, optimal control's behaviors in high-order problems lack of precision characterization, even where the existence of chattering phenomena remain unknown and overlooked. This paper establishes a theoretical framework of chattering phenomena in the problem, focusing on the uniqueness of state constraints inducing chattering, the upper bound of switching times in an unconstrained arc during chattering, and the convergence of states and costates to the chattering limit point. For the first time, this paper proves the existence of chattering phenomena in the problems. The chattering optimal control for 4th order problems with velocity constraints is precisely solved, providing an approach to plan strictly time-optimal snap-limited trajectories, while other cases of order n≤4 are proved to not allow chattering. The conclusions correct the longstanding misconception in the industry regarding the time-optimality of S-shaped trajectories with minimal switching times.