  This paper works towards merging two popular approaches in the control
community for safety control: Hamilton-Jacobi (HJ) reachability analysis and
Control Barrier Functions (CBFs). HJ Reachability has methods for direct
construction of value functions that provide safety guarantees and safe
controllers, however the online implementation can be overly conservative
and/or rely on jumpy bang-bang control. The CBF community has methods for
safe-guarding controllers in the form of point-wise optimization programs such
as CBF-QP, where the CBF-based safety certificate is used as a constraint.
However, finding a valid CBF for a general dynamical system is challenging.
This paper merges these two methods by introducing a new reachability
formulation inspired by the structure of CBFs that constructs a Control
Barrier-Value Function (CBVF). We verify that CBVF is a viscosity solution to a
novel Hamilton-Jacobi-Isaacs Variational Inequality and preserves the same
safety guarantee as the original reachability formulation. Finally, we propose
an online Quadratic Program formulation whose solution is always an optimal
control signal, which is inspired by the CBF-QP. We demonstrate the benefit of
using the CBVFs for double-integrator and Dubins car systems by comparing it to
previous methods.
