Predefined-time reinforcement learning for optimal feedback control

In this paper, we develop an online predefined time-convergent reinforcement learning architecture to solve the optimal predefined-time stabilization problem. Specifically, we introduce the problem of optimal predefined-time stabilization to construct feedback controllers that guarantee closed-loop system predefined-time stability while optimizing a given performance measure. The predefined time stability of the closed-loop system is established via a Lyapunov function satisfying a differential inequality while simultaneously serving as a solution to the steady-state Hamilton–Jacobi–Bellman equation ensuring optimality. Given that the Hamilton–Jacobi–Bellman equation is generally difficult to solve, we develop a critic-only reinforcement learning-based algorithm to learn the solution to the steady-state Hamilton–Jacobi–Bellman equation in predefined time. In particular, a non-Lipschitz experience replay-based learning law utilizing recorded and current data is introduced for updating the critic weights to learn the value function. The non-Lipschitz property of the dynamics and letting the learning rate as a function of the predefined time gives rise to predefined-time convergence, while the experience replay-based approach eliminates the need to satisfy the persistence of excitation condition as long as the recorded data set is sufficiently rich. Finally, an illustrative numerical example is provided to demonstrate the efficacy of the proposed approach.