Literature

The topic of verification is vast. While we only reserved a single week/chapter/block for it, it would easily fill a dedicated course, supported by a couple of books. Having a smaller time budget, we can still find some confirmation of usefulness even of a modest introduction in the Chapter 3 of [1]. Although we do not follow the book closely, we do cover some of their topics.

Among general references for hybrid system verification, we can recommend [2] for an overview. Although the book is not freely available for download, its web page contains quite some additional material such as slides and codes.

Reachability analysis (by set propagation techniques)

The overview paper [3] is recommendable. In addition, the manual for the CORA toolbox for Matlab althoffCORA2025Manual2025? (by the same author and his team) can do a good tutorial job.

Barier certificates

We based our introduction on [4], including the example. But a wealth of papers have been published on the topic, including the extension from analysis to control design in the form of control barrier functions, which has been introduced in [5]. A recent overview is in the book [6].

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References

[1]
H. Lin and P. J. Antsaklis, Hybrid Dynamical Systems: Fundamentals and Methods. in Advanced Textbooks in Control and Signal Processing. Cham: Springer, 2022. Accessed: Jul. 09, 2022. [Online]. Available: https://doi.org/10.1007/978-3-030-78731-8
[2]
S. Mitra, Verifying Cyber-Physical Systems: A Path to Safe Autonomy. in Cyber Physical Systems Series. Cambridge, MA, USA: MIT Press, 2021. Available: https://sayanmitracode.github.io/cpsbooksite/about.html
[3]
M. Althoff, G. Frehse, and A. Girard, “Set Propagation Techniques for Reachability Analysis,” Annual Review of Control, Robotics, and Autonomous Systems, vol. 4, no. 1, pp. 369–395, 2021, doi: 10.1146/annurev-control-071420-081941.
[4]
S. Prajna and A. Jadbabaie, “Safety Verification of Hybrid Systems Using Barrier Certificates,” in Hybrid Systems: Computation and Control, R. Alur and G. J. Pappas, Eds., in Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 2004, pp. 477–492. doi: 10.1007/978-3-540-24743-2_32.
[5]
P. Wieland and F. Allgöwer, “Constructive safety using control barrier functions,” in IFAC Proceedings Volumes, in 7th IFAC Symposium on Nonlinear Control Systems, vol. 40. Jan. 2007, pp. 462–467. doi: 10.3182/20070822-3-ZA-2920.00076.
[6]
W. Xiao, C. G. Cassandras, and C. Belta, Safe Autonomy with Control Barrier Functions: Theory and Applications. 2023.