Verification and Synthesis of Discrete-Time Control Barrier Functions

Erfan Shakhesi; W. P. M. H. Heemels; Alexander Katriniok

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arXiv · 2509.18685

Verification and Synthesis of Discrete-Time Control Barrier Functions

Erfan Shakhesi·W. P. M. H. Heemels·Alexander Katriniok

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Abstract

Discrete-time Control Barrier Functions (DTCBFs) have recently attracted interest for guaranteeing safety and synthesizing safe controllers for discrete-time dynamical systems. This paper addresses the open challenges of verifying candidate DTCBFs and synthesizing DTCBFs for general nonlinear discrete-time systems with input constraints and arbitrary safe sets. In particular, we propose a branch-and-bound method, inspired by the $\alpha$BB algorithm, for the verification of candidate DTCBFs in both cases, whether a corresponding control policy is known or unknown. We prove that this method, in a finite number of iterations, either verifies a given candidate function as a valid DTCBF or falsifies it by providing a counterexample (within predefined tolerances). As a second main contribution, we propose a novel bilevel optimization approach to synthesize a DTCBF and a corresponding control policy in finite time. This involves determining the unknown coefficients of a parameterized DTCBF and a parameterized control policy. Furthermore, we introduce various strategies to reduce the computational burden of the bilevel approach. We also demonstrate our methods using numerical case studies.

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BibTeX RIS

Erfan Shakhesi, W. P. M. H. Heemels, Alexander Katriniok. 2025-09-23. Verification and Synthesis of Discrete-Time Control Barrier Functions. https://arxiv.org/abs/2509.18685

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