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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

French

Dans ce document, nous tirons parti de la théorie et du cadre de l'approximation générale du bracket Lie d'ordre supérieur récemment développé pour les systèmes de contrôle-affinés pour montrer que nous pouvons concevoir des systèmes de recherche extrême (ES) qui combinent des flux basés sur: (i) l'approximation du bracket Lie de premier ordre, qui fournissent un taux de convergence exponentiel pour les fonctions objectives carrées et le taux de convergence polynomiale pour les fonctions objectives non carrées; et (ii) l'approximation du bracket Lie d'ordre supérieur, qui fournissent un taux de convergence exponentiel près du minimizateur même si la fonction objective est non-quadratique (c'est-à-dire,

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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