Finite Boundary-Layer Residence Certificates for Non-Strict Control Barrier Functions

Abstract

Non-strict control barrier function (CBF) conditions guarantee safety through forward invariance, but they do not preclude trajectories from remaining near the safe-set boundary for extended continuous time intervals. This paper develops a finite boundary-layer residence certificate for such settings. The certificate preserves the standard non-strict CBF safety condition and uses a bounded auxiliary function whose derivative is bounded away from zero in a prescribed boundary layer, yielding an explicit upper bound on every uninterrupted residence interval. For control-affine systems, a selected auxiliary branch is implemented as an additional affine constraint in a CBF-QP, and a tangential-input compatibility condition is given to ensure simultaneous feasibility with the hard CBF constraint for unconstrained inputs. A local-chart version handles angular or multi-valued auxiliary functions such as $atan2$. Single-integrator, double-integrator, and nonholonomic unicycle examples illustrate the resulting radial--tangential construction and its local-chart and feasibility limitations.