  Stability and safety are two important aspects in safety-critical control of
dynamical systems. It has been a well established fact in control theory that
stability properties can be characterized by Lyapunov functions. Reachability
properties can also be naturally captured by Lyapunov functions for finite-time
stability. Motivated by safety-critical control applications, such as in
autonomous systems and robotics, there has been a recent surge of interests in
characterizing safety properties using barrier functions. Lyapunov and barrier
functions conditions, however, are sometimes viewed as competing objectives. In
this paper, we provide a unified theoretical treatment of Lyapunov and barrier
functions in terms of converse theorems for stability properties with safety
guarantees and reach-avoid-stay type specifications. We show that if a system
(modeled as a dynamical system with measurable perturbations) possesses a
stability with safety property, then there exists a smooth Lyapunov function to
certify such a property. This Lyapunov function is shown to be defined on the
entire set of initial conditions from which solutions satisfy this property. A
similar but slightly weaker statement is made for reach-avoid-stay
specifications. We show by a simple example that the latter statement cannot be
strengthened without additional assumptions. We further extend the results for
systems with control inputs and prove existence of converse Lyapunov-barrier
functions for reach-and-avoid specifications. While the converse
Lyapunov-barrier theorems are not constructive, as with classical converse
Lyapunov theorems, we believe that the unified necessary and sufficient
conditions with a single Lyapunov-barrier function are of theoretical interest
and can hopefully shed some light on computational approaches.
