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# Reachability Analysis-based Safety-Critical Control using Online Fixed-Time Reinforcement Learning

*Proceedings of The 5th Annual Learning for Dynamics and Control Conference*, PMLR 211:1257-1270, 2023.

#### Abstract

In this paper, we address a safety-critical control problem using reachability analysis and design a reinforcement learning-based mechanism for learning online and in fixed-time the solution to the safety-critical control problem. Safety is assured by determining a set of states for which there does not exist an admissible control law generating a system trajectory reaching a set of forbidden states at a user-prescribed time instant. Specifically, we cast our safety-critical problem as a Mayer optimal feedback control problem whose solution satisfies the Hamilton-Jacobi-Bellman (HJB) equation and characterizes the set of safe states. Since the HJB equation is generally difficult to solve, we develop an online critic-only reinforcement learning-based algorithm for simultaneously learning the solution to the HJB equation and the safe set in fixed time. In particular, we introduce a non-Lipschitz experience replay-based learning law utilizing recorded and current data for updating the critic weights to learn the value function and the safe set. The non-Lipschitz property of the dynamics gives rise to fixed-time convergence, whereas the experience replay-based approach eliminates the need of satisfying the persistence of excitation condition provided that the recorded data is sufficiently rich. Simulation results illustrate the efficacy of the proposed approach.