Departmental Papers (ESE)


This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method.

Document Type

Government Publication

Subject Area


Date of this Version

August 2007


Copyright 2007 IEEE. Reprinted from IEEE Transactions on Automatic Control, Volume 52, Issue 8, August 2007, pages 1415-1428.

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Barrier certificates, hybrid systems, nonlinear systems, safety verification, stochastic systems, sum of squares optimization



Date Posted: 26 October 2007

This document has been peer reviewed.