Safe Physics-Informed Machine Learning for Optimal Predefined-Time Stabilization: A Lyapunov-Based Approach

In this article, we introduce the notion of safe predefined-time stability and address an optimal safe predefined-time stabilization problem. In particular, safe predefined-time stability characterizes parameter-dependent nonlinear dynamical systems whose trajectories starting in a given set of admissible states remain in the set of admissible states for all time and converge to an equilibrium point in a predefined time. Furthermore, we provide a Lyapunov theorem establishing sufficient conditions for safe predefined-time stability. We address the optimal safe predefined-time stabilization problem by synthesizing feedback controllers that guarantee closed-loop system safe predefined-time stability while optimizing a given performance measure. Specifically, safe predefined-time stability of the closed-loop system is guaranteed via a Lyapunov function satisfying a differential inequality while simultaneously serving as a solution to the steady-state Hamilton-Jacobi-Bellman (HJB) equation ensuring optimality. Given that the HJB equation is generally difficult to solve, we develop a physics-informed machine learning-based algorithm for learning the safely predefined-time stabilizing solution to the steady-state HJB equation. Several simulation results are provided to demonstrate the efficacy of the proposed approach.