  Motivated by the key role of control barrier functions (CBFs) in assessing
safety and enabling the synthesis of safe controllers in nonlinear control
systems, this paper presents a suite of converse results on CBFs. Given any
safe set, we first identify a set of general sufficient conditions which
guarantee the existence of a CBF. Our technical analysis also enables us to
define an extended notion of CBF which is always guaranteed to exist if the set
is safe. We next turn our attention to the problem of joint safety and
stability, and give conditions under which the notions of control
Lyapunov-barrier function (CLBF) and compatible control Lyapunov function (CLF)
and CBF pair are guaranteed to exist. Finally, we identify conditions under
which a CLBF and a compatible CLF-CBF pair can be constructed from a
non-compatible CLF-CBF pair. Throughout the paper, we intersperse different
examples and counterexamples to motivate our results and position them within
the state of the art.
