# Safety-Critical Control

DeepReach replaces the grid in Hamilton–Jacobi reachability with a neural value function but penalizes both sides of the residual equally, so the learned value can sit above the true one and unsafe states get labeled safe. ConDR instead trains toward a subsolution of the HJI variational inequality: the obstacle branch and terminal condition hold by construction, a one-sided loss pushes the differential residual below a safety margin, and a complementarity term blocks the degenerate all-unsafe solution. On Air3D, Air6D, a 9D multi-vehicle system and a 13D quadrotor, ConDR reports the lowest false-negative rate and the lowest closed-loop collision rates, with at most a small rise in intervention.

DeepReach replaces the grid in Hamilton–Jacobi reachability with a neural value function but penalizes both sides of the residual equally, so the learned value can sit above the true one and unsafe states get labeled safe. ConDR instead trains toward a subsolution of the HJI variational inequality: the obstacle branch and terminal condition hold by construction, a one-sided loss pushes the differential residual below a safety margin, and a complementarity term blocks the degenerate all-unsafe solution. On Air3D, Air6D, a 9D multi-vehicle system and a 13D quadrotor, ConDR reports the lowest false-negative rate and the lowest closed-loop collision rates, with at most a small rise in intervention.

Learned controllers for safety-critical systems usually maximize average reward first and add safety through penalties, constraints or runtime filters. CeDL instead trains without reward: it combines Lyapunov stability, control-barrier safety, temporal-logic robustness and feasibility margins into one certificate margin, and an adversarial falsifier repeatedly surfaces the scenarios closest to violation. Across simulated URLLC wireless resource allocation, energy-safe sensor-network routing and robotic obstacle avoidance, CeDL reports zero observed test-set violations and larger minimum certificate margins than reward-based RL, constrained RL, model predictive control, shielded RL and classical controllers.

Learned controllers for safety-critical systems usually maximize average reward first and add safety through penalties, constraints or runtime filters. CeDL instead trains without reward: it combines Lyapunov stability, control-barrier safety, temporal-logic robustness and feasibility margins into one certificate margin, and an adversarial falsifier repeatedly surfaces the scenarios closest to violation. Across simulated URLLC wireless resource allocation, energy-safe sensor-network routing and robotic obstacle avoidance, CeDL reports zero observed test-set violations and larger minimum certificate margins than reward-based RL, constrained RL, model predictive control, shielded RL and classical controllers.

This paper presents the composite adaptive control barrier function (CaCBF) algorithm for nonlinear control-affine systems with linear parametric uncertainty, deriving the adaptation law from a composite energy function that integrates a logarithmic safety barrier, a control Lyapunov function, and a parameter-error term; it proves forward invariance of the safe set for all bounded parameters without persistence of excitation, robustness of the safety guarantee to bounded state-derivative measurement errors, uniform ultimate boundedness of all closed-loop signals, and that the CaCBF admissible control set always contains the robust counterpart as a subset; simulations of adaptive cruise control, an omnidirectional robot, and a planar drone traversing a narrow gate show CaCBF recovers the pe

This paper presents the composite adaptive control barrier function (CaCBF) algorithm for nonlinear control-affine systems with linear parametric uncertainty, deriving the adaptation law from a composite energy function that integrates a logarithmic safety barrier, a control Lyapunov function, and a parameter-error term; it proves forward invariance of the safe set for all bounded parameters without persistence of excitation, robustness of the safety guarantee to bounded state-derivative measurement errors, uniform ultimate boundedness of all closed-loop signals, and that the CaCBF admissible control set always contains the robust counterpart as a subset; simulations of adaptive cruise control, an omnidirectional robot, and a planar drone traversing a narrow gate show CaCBF recovers the pe