2022/01/31 by Mohamed Maghenem, Maghenem, Mohamed, Ricardo. G. Sanfelice +1 · 2 citations
Decision Sciences · Engineering · #Advanced Control Systems Optimization #FOS: Mathematics #Fault Detection and Control Systems #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2201.13107
openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper presents converse theorems for safety in terms of barrier functions for unconstrained continuous-time systems modeled as differential inclusions. Via a counterexample, we show the lack of existence of autonomous and continuous barrier functions certifying safety for a nonlinear system that is not only safe but also has a smooth right-hand side. Guided by converse Lyapunov theorems for (non-asymptotic) stability,time-varying barrier functions and appropriate infinitesimal conditions are shown to be both necessary as well as sufficient under mild regularity conditions on the right-hand side of the system. More precisely, we propose a general construction of a time-varying barrier function in terms of a marginal function involving the finite-horizon reachable set. Using techniques from set-valued and nonsmooth analysis, we show that such a function guarantees safety when the system is safe. Furthermore, we show that the proposed barrier function construction inherits the regularity properties of the proposed reachable set. In addition, when the system is safe and smooth, we build upon the constructed barrier function to show the existence of a smooth barrier function guaranteeing safety. Comparisons and relationships to results in the literature are also presented.