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Combining Exponential and Polynomial Convergence Rates for Accelerated Extremum Seeking: A High-Order Lie Bracket Approach

Victoria Grushkovskaya, Sameh Abdelwahab Nasr Eisa · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.27239 · cited by 0 works (OpenAlex) · open-access PDF

Japanese

この論文では、我々は、最近開発された一般的な高い順位のLieブラケットアプローチ理論とコントロールアフィンシステムのフレームワークを用いて、我々は、(i)最初の順位のLieブラケットアプローチに基づいて流れを組み合わせる極端な検索(ES)システムを設計することができることを示す: (i)最初の順位のLieブラケットアプローチを提供する。

Machine translation (Meta M2M100); the English original follows.

Abstract

In this paper, we leverage the recently developed general higher-order Lie bracket approximation theory and framework for control-affine systems to show that we can design extremum seeking (ES) systems that combine flows based on: (i) first-order Lie bracket approximation, which provide an exponential convergence rate for quadratic objective functions and polynomial convergence rate for non-quadratic objective functions; and (ii) higher-order Lie bracket approximation, which provide exponential convergence rate near the minimizer even if the objective function is non-quadratic (i.e., flatter and possesses vanishing Hessian at the minimizer). Such combination provides an accelerated ES design that benefits from the polynomial decay rate farther away from the minimizer and exponential decay rate near the minimizer even for complicated nature of objective functions. We provide numerical simulations to demonstrate the results, including multi-variable and nonpolynomial flat-bottomed objective functions.

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Topics: Mathematics, Hessian matrix, Rate of convergence, Polynomial, Exponential function, Applied mathematics

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