As autonomous systems become more ubiquitous in daily life, ensuring high performance with guaranteed safety is crucial. However, safety and performance could be competing objectives, which makes their co-optimization difficult. Learning-based methods, such as Constrained Reinforcement Learning (CRL), achieve strong performance but lack formal safety guarantees due to safety being enforced as soft constraints, limiting their use in safety-critical settings. Conversely, formal methods such as Hamilton-Jacobi (HJ) Reachability Analysis and Control Barrier Functions (CBFs) provide rigorous safety assurances but often neglect performance, resulting in overly conservative controllers. To bridge this gap, we formulate the co-optimization of safety and performance as a state-constrained optimal control problem, where performance objectives are encoded via a cost function and safety requirements are imposed as state constraints. We demonstrate that the resultant value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation, which we approximate efficiently using a novel physics-informed machine learning framework. In addition, we introduce a conformal prediction-based verification strategy to quantify the learning errors, recovering a high-confidence safety value function, along with a probabilistic error bound on performance degradation. Through several case studies, we demonstrate the efficacy of the proposed framework in enabling scalable learning of safe and performant controllers for complex, high-dimensional autonomous systems.

翻译：随着自主系统在日常生活中日益普及，确保其高性能与可验证的安全性变得至关重要。然而，安全性与性能目标可能存在冲突，使得二者的协同优化变得困难。基于学习的方法，如约束强化学习（CRL），虽能实现较强的性能，但由于安全仅作为软约束实施，缺乏形式化的安全保证，限制了其在安全关键场景中的应用。反之，形式化方法，如 Hamilton-Jacobi（HJ）可达性分析与控制屏障函数（CBFs），提供了严格的安全保证，但往往忽略性能，导致控制器过于保守。为弥合这一差距，我们将安全与性能的协同优化表述为一个状态约束的最优控制问题，其中性能目标通过代价函数编码，安全要求则作为状态约束施加。我们证明了所得值函数满足 Hamilton-Jacobi-Bellman（HJB）方程，并采用一种新颖的基于物理信息的机器学习框架对其进行高效近似。此外，我们引入了一种基于共形预测的验证策略，以量化学习误差，从而恢复一个高置信度的安全值函数，并给出性能退化的概率误差界。通过多个案例研究，我们证明了所提框架在实现复杂、高维自主系统的安全与高性能控制器的可扩展学习方面的有效性。