Tis paper introduces an inclusive class of fxed-time stable continuous-time gradient fows (GFs). Tis class of GFs is then leveraged to learn optimal control solutions for nonlinear systems in fxed time. It is shown that the presented GF guarantees convergence within a fxed time from any initial condition to the exact minimum of functions that satisfy the Polyak–Łojasiewicz (PL) inequality. Te presented fxed-time GF is then utilized to design fxed-time optimal adaptive control algorithms. To this end, a fxed-time reinforcement learning (RL) algorithm is developed on the basis of a single network adaptive critic (SNAC) to learn the solution to an infnite-horizon optimal control problem in a fxed-time convergent, online, adaptive, and forward-in-time manner. It is shown that the PL inequality in the presented RL algorithm amounts to a mild inequality condition on a few collected samples. Tis condition is much weaker than the standard persistence of excitation (PE) and fnite duration PE that relies on a rank condition of a dataset. Tis is crucial for learning-enabled control systems as control systems can commit to learning an optimal controller from the beginning, in sharp contrast to existing results that rely on the PE and rank condition, and can only commit to learning after rich data samples are collected. Simulation results are provided to validate the performance and efcacy of the presented fxed-time RL algorithm

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