Barrier certificate

This article is about the topic in dynamical systems. For the topic in constrained optimization, see barrier function.

A barrier certificate[1] or barrier function is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system.[2] That is, a barrier function can be used to show that if a solution starts in a given set, then it cannot leave that set.

Showing that a set is forward invariant is an aspect of safety, which is the property where a system is guaranteed to avoid obstacles specified as an unsafe set.

Barrier certificates play the analogical role for safety to the role of Lyapunov functions for stability. For every ordinary differential equation that robustly fulfills a safety property of a certain type there is a corresponding barrier certificate.[3]

  1. ^ Prajna, Stephen, and Ali Jadbabaie. "Safety verification of hybrid systems using barrier certificates." International Workshop on Hybrid Systems: Computation and Control. Springer, Berlin, Heidelberg, 2004.
  2. ^ Maghenem, M., Sanfelice, R. G. (February 2021). "Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions". Automatica. 124: 109328. arXiv:1908.03980. doi:10.1016/j.automatica.2020.109328. ISSN 0005-1098.
  3. ^ Stefan Ratschan: "Converse Theorems for Safety and Barrier Certificates". IEEE Trans. on Automatic Control, Volume 63, Issue 8, 2018