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1. Control barrier functionals: Safety‐critical control for time delay systems

   https://doi.org/10.1002/rnc.6751  

   Kiss, Adam K.; Molnar, Tamas G.; Ames, Aaron D.; Orosz, Gabor (August 2023, International Journal of Robust and Nonlinear Control)

   Abstarct This work presents a theoretical framework for the safety‐critical control of time delay systems. The theory of control barrier functions, that provides formal safety guarantees for delay‐free systems, is extended to systems with state delay. The notion of control barrier functionals is introduced, to attain formal safety guarantees by enforcing the forward invariance of safe sets defined in the infinite dimensional state space. The proposed framework is able to handle multiple delays and distributed delays both in the dynamics and in the safety condition, and provides an affine constraint on the control input that yields provable safety. This constraint can be incorporated into optimization problems to synthesize pointwise optimal and provable safe controllers. The applicability of the proposed method is demonstrated by numerical simulation examples. 

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2. Safe Online Convex Optimization with Unknown Linear Safety Constraints

   Chaudhary Sapana; Kalathil, Dileep (February 2022, AAAI Conference on Artificial Intelligence (AAAI))

   We study the problem of safe online convex optimization, where the action at each time step must satisfy a set of linear safety constraints. The goal is to select a sequence of ac- tions to minimize the regret without violating the safety constraints at any time step (with high probability). The parameters that specify the linear safety constraints are unknown to the algorithm. The algorithm has access to only the noisy observations of constraints for the chosen actions. We pro- pose an algorithm, called the Safe Online Projected Gradient Descent(SO-PGD) algorithm to address this problem. We show that, under the assumption of the availability of a safe baseline action, the SO-PGD algorithm achieves a regret O(T^2/3). While there are many algorithms for online convex optimization (OCO) problems with safety constraints avail- able in the literature, they allow constraint violations during learning/optimization, and the focus has been on characterizing the cumulative constraint violations. To the best of our knowledge, ours is the first work that provides an algorithm with provable guarantees on the regret, without violating the linear safety constraints (with high probability) at any time step. 

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3. Finding Interior Optimum of Black-box Constrained Objective with Bayesian Optimization

   Zhang, F; Chen, Y (July 2025, The 41st Conference on Uncertainty in Artificial Intelligence)

   Optimizing objectives under constraints, where both the objectives and constraints are black box functions, is a common scenario in real-world applications such as the design of medical therapies, industrial process optimization, and hyperparameter optimization. One popular approach to handle these complex scenarios is Bayesian Optimization (BO). However, when it comes to the theoretical understanding of constrained Bayesian optimization (CBO), the existing framework often relies on heuristics, approximations, or relaxation of objectives and, therefore, lacks the same level of theoretical guarantees as in canonical BO. In this paper, we exclude the boundary candidates that could be compromised by noise perturbation and aim to find the interior optimum of the black-box-constrained objective. We rely on the insight that optimizing the objective and learning the constraints can both help identify the high-confidence regions of interest (ROI) that potentially contain the interior optimum. We propose an efficient CBO framework that intersects the ROIs identified from each aspect on a discretized search space to determine the general ROI. Then, on the ROI, we optimize the acquisition functions, balancing the learning of the constraints and the optimization of the objective. We showcase the efficiency and robustness of our proposed CBO framework through the high probability regret bounds for the algorithm and extensive empirical evidence. 

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4. Model Predictive Control Barrier Functions: Guaranteed Safety with Reduced Conservatism and Shortened Horizon

   https://doi.org/10.23919/ACC60939.2024.10644741  

   Abdi, Hossein; Zhao, Pan; Hovakimyan, Naira; Ghabcheloo, Reza (July 2024, IEEE)

   In this study, we address the problem of safe control in systems subject to state and input constraints by integrating the Control Barrier Function (CBF) into the Model Predictive Control (MPC) formulation. While CBF offers a conservative policy and traditional MPC lacks the safety guarantee beyond the finite horizon, the proposed scheme takes advantage of both MPC and CBF approaches to provide a guaranteed safe control policy with reduced conservatism and a shortened horizon. The proposed methodology leverages the sum-of-square (SOS) technique to construct CBFs that make forward invariant safe sets in the state space that are then used as a terminal constraint on the last predicted state. CBF invariant sets cover the state space around system fixed points. These islands of forward invariant CBF sets will be connected to each other using MPC. To do this, we proposed a technique to handle the MPC optimization problem subject to the combination of intersections and union of constraints. Our approach, termed Model Predictive Control Barrier Functions (MPCBF), is validated using numerical examples to demonstrate its efficacy, showing improved performance compared to classical MPC and CBF. 

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5. Safe Physics-informed Machine Learning for Dynamics and Control

   Drgona, Jan; Nghiem, Truong X; Beckers, Thomas; Fazlyab, Mahyar; Mallada, Enrique; Jones, Colin; Vrabie, Draguna; Brunton, Steven L; Findeisen, Rolf (July 2025, IEEE)

   This tutorial paper focuses on safe physics- informed machine learning in the context of dynamics and control, providing a comprehensive overview of how to integrate physical models and safety guarantees. As machine learning techniques enhance the modeling and control of complex dynamical systems, ensuring safety and stability remains a critical challenge, especially in safety-critical applications like autonomous vehicles, robotics, medical decision-making, and energy systems. We explore various approaches for embedding and ensuring safety constraints, including structural priors, Lyapunov and Control Barrier Functions, predictive control, projections, and robust optimization techniques. Additionally, we delve into methods for uncertainty quantification and safety verification, including reachability analysis and neural network verification tools, which help validate that control policies remain within safe operating bounds even in uncertain environments. The paper includes illustrative examples demonstrating the implementation aspects of safe learning frameworks that combine the strengths of data-driven approaches with the rigor of physical principles, offering a path toward the safe control of complex dynamical systems. 

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