HJB-based online safety-embedded critic learning for uncertain systems with self-triggered mechanism

This paper presents a learning-based optimal control framework for safety-critical systems with parametric uncertainties, addressing both time-triggered and self-triggered controller implementations. First, we develop a robust control barrier function (RCBF) incorporating Lyapunov-based compensation terms to rigorously guarantee safety despite parametric uncertainties. Building on this safety guarantee, we formulate the constrained optimal control problem as the minimization of a novel safety-embedded value function, where the RCBF is involved via a Lagrange multiplier that adaptively balances safety constraints against optimal stabilization objectives. To enhance computational efficiency, we propose a self-triggered implementation mechanism that reduces control updates while maintaining dual stability-safety guarantees. The resulting self-triggered constrained Hamilton-Jacobi-Bellman (HJB) equation is solved through an online safety-embedded critic learning framework, with the Lagrange multiplier computed in real time to ensure safety. Numerical simulations demonstrate the effectiveness of the proposed approach in achieving both safety and control performance.