Energy-Aware Bayesian Control Barrier Functions for Physics-Informed Gaussian Process Dynamics

We study safe control for dynamical systems whose continuous-time dynamics are learned with Gaussian processes (GPs), focusing on mechanical and port-Hamiltonian systems where safety is naturally expressed via energy constraints. The availability of a GP Hamiltonian posterior naturally raises the question of how to systematically exploit this structure to design an energy-aware control barrier function with high-probability safety guarantees. We address this problem by developing a Bayesian-CBF framework and instantiating it with energy-aware Bayesian-CBFs (EB-CBFs) that construct conservative energy-based barriers directly from the Hamiltonian and vector-field posteriors, yielding safety filters that minimally modify a nominal controller while providing probabilistic energy safety guarantees. Numerical simulations on a mass-spring system demonstrate that the proposed EB-CBFs achieve high-probability safety under noisy sampled GP-learned dynamics.