# EECI 2012: Hybrid Systems Verification

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This lecture focuses on the verification of hybrid systems. We first discuss finite-state under- and over-approximations of hybrid dynamics and how these finite-state models coupled with the model checking tools can be used to verify temporal logic properties against hybrid dynamics. Then, we move to deductive verification and a computational procedure for constructing Lyapunov-type functions (e.g., barrier certificates) which witness the fact that a hybrid system satisfies certain temporal specifications. We finally introduce the notions of approximate bisimulations and bisimulation functions and how they can be used in verification of temporal properties.

## Lecture Materials

- Lecture slides: Hybrid Systems Verification

## Additional Material and Further Reading

Material on part III. Links to software packages, slides, and notes on quantitative analysis of nonlinear systems. Shorter version is in these slides

Stephen Prajna's dissertation on verifying temporal properties for hybrid dynamical systems.

Help on SOS: a paper on the very basics of sum-of-squares programming and their use in nonlinear system verification.

Minimizing Polynomial Functions by P. Parrilo and B. Sturmfels on global optimization of polynomial functions and Positivstellensatz (generalizations of the S-procedure).

Lecture Notes on Hybrid Systems (by John Lygeros): A rough introduction to hybrid systems. Chapters 5 and 6 are on various analysis techniques relevant for this short course.