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

Time-optimal control for high-order chain-of-integrator systems with full state constraints remains an open and challenging problem within the discipline of optimal control. The behavior of optimal control in high-order problems lacks precise characterization. Even the existence of the chattering phenomenon, i.e., the control switches infinitely many times over a finite period, remains unknown. This article establishes a theoretical framework for chattering in the problem, providing novel findings 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 article proves the existence of chattering in the considered problem. The chattering optimal control for fourth-order problems with velocity constraints is precisely solved, providing an approach to plan time-optimal snap-limited trajectories. Other cases of order $n\leq 4$ are proved not to allow chattering. The conclusions challenge a prevalent conception in the industry concerning the time-optimality of S-shaped trajectories with finite switching times.