Prescribed-Time Newton Extremum Seeking using Delays and Time-Periodic Gains

We study prescribed-time extremum seeking (PT-ES) for scalar maps in the presence of time delays. The PT-ES problem has been studied by Yilmaz and Krstic in 2023 using chirpy probing and time-varying gains that grow unbounded. To alleviate the gain singularity, in this paper we present an alternative approach, employing delays with bounded time-periodic gains, for achieving prescribed-time convergence to the extremum. Our results are not extensions or refinements of earlier works, but a new methodological direction --applicable even when the map has no delay. The main PT-ES algorithm compensates the map's delay and uses perturbation-based and the Newton (rather than gradient) approaches. With the help of averaging theorems in infinite dimension, specifically Retarded Functional Differential Equations (RFDEs), we conduct a prescribed-time convergence analysis on a suitable averaged target ES system, which contains the time-periodic gains of the map and feedback delays. We further extend our method to multivariable static maps and illustrate our results through numerical simulations.