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

Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specifications