Relaxed Control Barrier Functions with Slack: Feasible Safety Filtering with Explicit Violation Bounds

We address infeasibility of Control Barrier Function (CBF) constraints in multi-agent systems with conflicting safety requirements. Classical CBF-based safety filters rely on forward invariance, which breaks down when constraint sets become incompatible. We propose a relaxed CBF formulation with slack variables that guarantees feasibility at every time step. The resulting controller is implemented as a quadratic program and enforces safety constraints up to a bounded violation. Our main result establishes an explicit lower bound on the safety function for discrete-time systems with bounded disturbance. The bound depends only on system parameters, slack limits, and disturbance magnitude. Theoretical result: h (k+1) ≥ (1 − αΔt) h (k) − w (k) which yields: hₘin ≥ −δₑff Key contributions: – Always-feasible safety filtering via slack variables – Explicit, trajectory-independent safety bound – Quantitative trade-off between feasibility and safety violation – Validation in multi-agent collision avoidance scenarios This work transforms safety from a strict invariance property into a bounded and always-feasible control objective.