# EECI 2012: Deductive Verification of Hybrid Systems

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This lecture focuses on the verification of hybrid systems using deductive (theorem proving) methods. We describe hybrid systems that combine continuous and discrete states. We then describe two methods for deductive verification. First, we discuss a computational procedure for constructing Lyapunov-type functions (e.g., barrier certificates) that witness the fact that a hybrid system satisfies certain temporal specifications. Second, we consider control protocols for cooperation and decision making in multi-agent systems, as illustrated by the RoboFlag example introduced in the first lecture. We show how to implement a simple protocol for distributed target assignment in a simplified version of the problem (the "RoboFlag drill") and prove stability of the protocol.

## Lecture Materials

- Lecture slides: Deductive Verification of Hybrid Systems

Some of the slides are from a short course (Quantitative Local Analysis of Nonlinear Systems Using Sum-of-Squares Decompositions) by Balas, Packard, Seiler, and Topcu.

## Further Reading

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.

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).

A Computation and Control Language for Multi-Vehicle Systems, E. Klavins. Int’l Conference on Robotics and Automation, 2004.

Distributed Algorithms for Cooperative Control, E. Klavins and R. M. Murray.

*IEEE Pervasive Computing*, 2004.