Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. However, verifying that a learned CBF guarantees safety remains a challenging research problem. This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions.

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems.

In what follows, we give some details of content omitted in the paper due to space limit. The proof approach is based on Nagumo's Theorem, which gives necessary and sufficient conditions Definition 2. Let A be a closed set. The following is a fundamental preliminary result for establishing positive invariance. Proposition 2. F or any x 2 @ D, we have T Lemma 2 is a consequence of Proposition 2. For ease of exposition, we first reproduce the lemma First, suppose that condition (i) holds. Next, suppose that condition (ii) holds.

In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time.

Safety filters, particularly those based on control barrier functions, have gained increased interest as effective tools for safe control of dynamical systems. Existing correct-by-construction synthesis algorithms for such filters, however, suffer from the curse-of-dimensionality. Deep learning approaches have been proposed in recent years to address this challenge. In this paper, we add to this set of approaches an algorithm for training neural control barrier functions from offline datasets. Such functions can be used to design constraints for quadratic programs that are then used as safety filters. Our algorithm trains these functions so that the system is not only prevented from reaching unsafe states but is also disincentivized from reaching out-of-distribution ones, at which they would be less reliable. It is inspired by Conservative Q-learning, an offline reinforcement learning algorithm. We call its outputs Conservative Control Barrier Functions (CCBFs). Our empirical results demonstrate that CCBFs outperform existing methods in maintaining safety while minimally affecting task performance. Source code is available at https://github.com/tabz23/CCBF.

Control Barrier Functions (CBFs) are a practical approach for designing safety-critical controllers, but constructing them for arbitrary nonlinear dynamical systems remains a challenge. Recent efforts have explored learning-based methods, such as neural CBFs (NCBFs), to address this issue. However, ensuring the validity of NCBFs is difficult due to potential learning errors. In this letter, we propose a novel framework that leverages split-conformal prediction to generate formally verified neural CBFs with probabilistic guarantees based on a user-defined error rate, referred to as CP-NCBF. Unlike existing methods that impose Lipschitz constraints on neural CBF-leading to scalability limitations and overly conservative safe sets--our approach is sample-efficient, scalable, and results in less restrictive safety regions. We validate our framework through case studies on obstacle avoidance in autonomous driving and geo-fencing of aerial vehicles, demonstrating its ability to generate larger and less conservative safe sets compared to conventional techniques.

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems. State-of-the-art exact approaches to verifying safety of NCBF-based controllers exploit the piecewise-linear structure of ReLU neural networks, however, such approaches still rely on enumerating all of the activation regions of the network near the safety boundary, thus incurring high computation cost. In this paper, we propose a framework for Synthesis with Efficient Exact Verification (SEEV). Our framework consists of two components, namely (i) an NCBF synthesis algorithm that introduces a novel regularizer to reduce the number of activation regions at the safety boundary, and (ii) a verification algorithm that exploits tight over-approximations of the safety conditions to reduce the cost of verifying each piecewise-linear segment. Our simulations show that SEEV significantly improves verification efficiency while maintaining the CBF quality across various benchmark systems and neural network structures. Our code is available at https://github.com/HongchaoZhang-HZ/SEEV.