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 [4] (by the same author and his team) can do a good tutorial job.

Barier certificates

We based our introduction on [5], 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 [6]. A recent overview is in the book [7].

Temporal logics

Two popular monographs on verification (model checking) based on temporal logics such as LTL, CTL and CTL* are [8], and [9]. These do not cover hybrid systems, though. Still they are recommendable (at least their first chapters) for understanding the basics.

Some learning material and sketches of applications of temporal logics in control systems for robotics and autonomous driving are in the lectures [10], and [11].

A temporal logic particularly useful for specifying more complex requirements on (hybrid) control systems is Signal Temporal Logic (STL). Its treatment of this framework in textbooks is still rather sketchy. Research papers are then the only source. STL has been introduced in the readable [12]. Robustness degree has been described in [13] and the self-contained slides [14]. Some more recent papers that also contain other relevant references are [15], [16], [17], [18].

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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]
M. Althoff, N. Kochdumper, M. Wetzlinger, and T. Ladner, “CORA 2025 Manual.” 2025. Available: https://tumcps.github.io/CORA/data/archive/manual/Cora2025Manual.pdf
[5]
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.
[6]
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.
[7]
W. Xiao, C. G. Cassandras, and C. Belta, Safe Autonomy with Control Barrier Functions: Theory and Applications. 11 kv{\v e}tna 2023.
[8]
C. Baier and J.-P. Katoen, Principles of Model Checking. Cambridge, MA, USA: MIT Press, 2008. Available: https://mitpress.mit.edu/books/principles-model-checking
[9]
E. M. Clarke, Jr, O. Grumberg, D. Kroening, D. Peled, and H. Veith, Model Checking, 2nd ed. in Cyber Physical Systems Series. Cambridge, MA, USA: MIT Press, 2018. Available: https://mitpress.mit.edu/9780262038836/model-checking/
[10]
R. M. Murray, U. Topcu, and N. Wongpiromsarn, “Lecture 3 Linear Temporal Logic (LTL).” Belgrade (Serbia), Mar. 2020. Available: http://www.cds.caltech.edu/~murray/courses/eeci-sp2020/L3_ltl-09Mar2020.pdf
[11]
N. Wongpiromsarn, R. M. Murray, and U. Topcu, “Lecture 4 Model Checking and Logic Synthesis.” Belgrade (Serbia), Mar. 2020. Available: http://www.cds.caltech.edu/~murray/courses/eeci-sp2020//L4_model_checking-09Mar2020.pdf
[12]
O. Maler and D. Nickovic, “Monitoring Temporal Properties of Continuous Signals,” in Formal Techniques, Modelling and Analysis of Timed and Fault-Tolerant Systems, Y. Lakhnech and S. Yovine, Eds., Berlin, Heidelberg: Springer, 2004, pp. 152–166. doi: 10.1007/978-3-540-30206-3_12.
[13]
A. Donzé and O. Maler, “Robust Satisfaction of Temporal Logic over Real-Valued Signals,” in Formal Modeling and Analysis of Timed Systems, K. Chatterjee and T. A. Henzinger, Eds., Berlin, Heidelberg: Springer, 2010, pp. 92–106. doi: 10.1007/978-3-642-15297-9_9.
[14]
A. Donzé, “On Signal Temporal Logic.” University of California, Berkeley, Feb. 2014. Available: https://people.eecs.berkeley.edu/~sseshia/fmee/lectures/EECS294-98_Spring2014_STL_Lecture.pdf
[15]
P. Yu, Y. Gao, F. J. Jiang, K. H. Johansson, and D. V. Dimarogonas, “Online control synthesis for uncertain systems under signal temporal logic specifications,” The International Journal of Robotics Research, vol. 43, no. 6, pp. 765–790, May 2024, doi: 10.1177/02783649231212572.
[16]
V. Raman, M. Maasoumy, and A. Donzé, “Model predictive control from signal temporal logic specifications: A case study,” in Proceedings of the 4th ACM SIGBED International Workshop on Design, Modeling, and Evaluation of Cyber-Physical Systems, in CyPhy ’14. New York, NY, USA: Association for Computing Machinery, Apr. 2014, pp. 52–55. doi: 10.1145/2593458.2593472.
[17]
S. S. Farahani, V. Raman, and R. M. Murray, “Robust Model Predictive Control for Signal Temporal Logic Synthesis,” IFAC-PapersOnLine, vol. 48, no. 27, pp. 323–328, Jan. 2015, doi: 10.1016/j.ifacol.2015.11.195.
[18]
S. S. Farahani, R. Majumdar, V. S. Prabhu, and S. Soudjani, “Shrinking Horizon Model Predictive Control With Signal Temporal Logic Constraints Under Stochastic Disturbances,” IEEE Transactions on Automatic Control, vol. 64, no. 8, pp. 3324–3331, Aug. 2019, doi: 10.1109/TAC.2018.2880651.