# Control Barrier Function

For stochastic-gradient optimization with very many inequality constraints, an adaptive constraint-sampling method stops drawing each step's constraint uniformly from all constraints and instead maintains an "important set" of recently violated constraints, sampling from that set or from all constraints with fixed probabilities. On 100-dimensional synthetic convex problems it lowers the final Euclidean distance of the iterates to the constrained minimizer by about two orders of magnitude on average for ABSGD, Projection-SGD and Huber-penalty SGD, and by one to two orders for APDSG; in Dubins Car control-barrier-function synthesis it lowers maximum constraint violation by half an order to one order of magnitude.

For stochastic-gradient optimization with very many inequality constraints, an adaptive constraint-sampling method stops drawing each step's constraint uniformly from all constraints and instead maintains an "important set" of recently violated constraints, sampling from that set or from all constraints with fixed probabilities. On 100-dimensional synthetic convex problems it lowers the final Euclidean distance of the iterates to the constrained minimizer by about two orders of magnitude on average for ABSGD, Projection-SGD and Huber-penalty SGD, and by one to two orders for APDSG; in Dubins Car control-barrier-function synthesis it lowers maximum constraint violation by half an order to one order of magnitude.

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.

Swarm synchronization for green IoT no longer broadcasts state on a fixed cycle. ET-S3 ties each agent's trigger threshold to its remaining battery, so low-energy nodes transmit less, and a control-barrier-function quadratic program filters the consensus input for safety. In a simulation of 100 dynamic agents over 1000 synchronization cycles, cumulative payload was 3.35 MB, 18.2% of the 18.40 MB of periodic consensus, with a 0.0% safety violation rate and convergence in 138 steps.

Swarm synchronization for green IoT no longer broadcasts state on a fixed cycle. ET-S3 ties each agent's trigger threshold to its remaining battery, so low-energy nodes transmit less, and a control-barrier-function quadratic program filters the consensus input for safety. In a simulation of 100 dynamic agents over 1000 synchronization cycles, cumulative payload was 3.35 MB, 18.2% of the 18.40 MB of periodic consensus, with a 0.0% safety violation rate and convergence in 138 steps.

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