Cancellation of moving coefficients in curved-boundary classification

Determine whether the moving coefficients in the curved-boundary classification for continuous-time multiparametric optimal control can cancel when the boundary structure contains several arcs, including cases in which the boundary has a switching point.

Background

The paper classifies boundaries between critical regions as hyperplanes or curved surfaces based on whether the event time and arc durations remain fixed along the boundary. For boundaries involving several arcs, the curved classification relies on the assumption that the coefficients depending on moving event times or arc durations do not cancel. The paper establishes that this assumption can be removed when the boundary structure has no switching point, but leaves unresolved whether cancellations can occur in the more general several-arc setting and how such cancellations would affect the classification.

References

Two questions remain open. For several arcs the curved classification assumes that the moving coefficients do not cancel, an assumption Theorem~\ref{thm:main}(b) removes only when the boundary has no switching point; and pure state constraints, where the costate can jump, are outside the present class.

— Advances in Continuous-Time Multiparametric Optimal Control: The Role of Time-Varying Active Constraints  (2610.02680 - Lamakani et al., 2 Oct 2026) in Conclusion

Two questions remain open. For several arcs the curved classification assumes that the moving coefficients do not cancel, an assumption Theorem~\ref{thm:main}(b) removes only when the boundary has no switching point; and pure state constraints, where the costate can jump, are outside the present class.

— Advances in Continuous-Time Multiparametric Optimal Control: The Role of Time-Varying Active Constraints  (2610.02680 - Lamakani et al., 2 Oct 2026) in Conclusion