Solution properties of convex sweeping processes with velocity constraints

Abstract

Some properties of solutions of convex sweeping processes with velocity constraints are studied in this paper. Namely, the solution stability with respect to the initial value, the boundedness, the closedness, and the convexity of the solution set are discussed in detail. In addition, an outer estimate for the solution set is given here for the first time and two open questions are raised for further research. Our investigations complement the preceding ones on the solution existence and the solution uniqueness of convex sweeping processes with velocity constraints.

Keywords:

• Sweeping process
• velocity constraint
• solution property
• Lipschitz-like property
• inner continuity
• Bochner integration
• Sobolev space

AMS CLASSIFICATIONS:

• 49J40
• 47J20
• 47J22
• 58E35
• 34G25

Acknowledgments

The author would like to thank Prof. Samir Adly, Prof. Nguyen Khoa Son, and Prof. Nguyen Dong Yen for encouragement and valuable discussions on the subject. Valuable suggestions of the two anonymous referees, which have helped to significantly improve the paper presentation, are gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the project “Sweeping processes with velocity constraints” (Code: ICRTM03$\mathrm{\_}$2021.05) of the International Center for Research and Postgraduate Training in Mathematics (ICRTM) under the auspices of UNESCO of Institute of Mathematics, Vietnam Academy of Science and Technology.