Online Safety-Embedded Critic Learning for Uncertain Systems With Self-Triggered Mechanism

This paper addresses the problem of learning-based optimal control for safety-critical systems, where optimal stabilization, safety guarantees, and real-time implementability must be achieved simultaneously. First, to bridge optimal stabilization and hard safety constraints, we construct a primal—dual value function. This is enabled by a closed—form Lagrange multiplier induced by a Lyapunov-compensated control barrier function (LCBF), which embeds an explicit safety-filter structure and provides an analytic safety—stabilization trade-off in the optimal controller. Second, to enable efficient real-time execution, we develop a self-triggered (ST) mechanism based on dual stability—safety margins, which adaptively schedules controller updates and substantially reduces computation while preserving certified safety and stability. Third, to approximate the resulting LCBF-constrained optimal controller implemented via the ST mechanism, we propose an online safety-embedded critic learning framework with concurrent learning, which evaluates Bellman errors under the safety-certified controller to enable safe exploration toward a performance-driven policy. Simulation results verify that the proposed framework delivers safety assurance, reliable control performance, and reduced computational load.