Extremum Seeking Control: Three Revolutions and the Road Ahead
Abstract
The history of extremum seeking is not merely the history of an algorithm; it is the history of an idea. Few ideas in control engineering have demonstrated the remarkable longevity of extremum seeking control (ESC). Invented more than one hundred years ago, ESC has continually reinvented itself while remaining faithful to its original objective: enabling systems to optimize their performance without relying on accurate mathematical models. This perspective article proposes that the evolution of ESC is best understood through three scientific revolutions. The first Engineering Revolution (1922-1999) established the engineering principles of model-free optimization; the second Mathematical Revolution (2000-2010) provided the rigorous mathematical foundations that transformed ESC into a mature discipline of nonlinear control; and the third ongoing Infinite-Dimensional and Cyber-Physical Revolution (2010-present) continues to expand its scope toward delays, partial differential equations, distributed optimization, event-triggered implementations, and increasingly complex cyber-physical systems. Beyond recounting this historical evolution, we offer a personal perspective on why ESC has remained relevant across successive technological eras. We argue that its enduring influence arises because the fundamental engineering challenge has never changed: "how can a dynamical system learn to improve its own performance when the optimum is unknown?" As optimization, learning, and feedback control become increasingly intertwined, ESC appears uniquely positioned to contribute to a new generation of intelligent autonomous systems, pointing toward what may become the field's fourth scientific revolution.
Related
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- Resilient Extremum Seeking Control for Cyber-Physical Systems Under Denial-of-Service Attacks
- Combining Exponential and Polynomial Convergence Rates for Accelerated Extremum Seeking: A High-Order Lie Bracket Approach
- Multivariable Extremum Seeking for Locally Lipschitz Objectives
- Distributed Stochastic Extremum Seeking for Optimal Resource Allocation over Networks with Time Delays
BibTeX
@misc{oliveira2026extremum,
title = {{Extremum Seeking Control: Three Revolutions and the Road Ahead}},
author = {Tiago Roux Oliveira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00484},
eprint = {2609.00484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00484},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extremum Seeking Control: Three Revolutions and the Road Ahead
AU - Tiago Roux Oliveira
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00484
UR - https://arxiv.org/abs/2609.00484
AB - The history of extremum seeking is not merely the history of an algorithm; it is the history of an idea. Few ideas in control engineering have demonstrated the remarkable longevity of extremum seeking control (ESC). Invented more than one hundred years ago, ESC has continually reinvented itself while remaining faithful to its original objective: enabling systems to optimize their performance without relying on accurate mathematical models. This perspective article proposes that the evolution of ESC is best understood through three scientific revolutions. The first Engineering Revolution (1922-1999) established the engineering principles of model-free optimization; the second Mathematical Revolution (2000-2010) provided the rigorous mathematical foundations that transformed ESC into a mature discipline of nonlinear control; and the third ongoing Infinite-Dimensional and Cyber-Physical Revolution (2010-present) continues to expand its scope toward delays, partial differential equations, distributed optimization, event-triggered implementations, and increasingly complex cyber-physical systems. Beyond recounting this historical evolution, we offer a personal perspective on why ESC has remained relevant across successive technological eras. We argue that its enduring influence arises because the fundamental engineering challenge has never changed: "how can a dynamical system learn to improve its own performance when the optimum is unknown?" As optimization, learning, and feedback control become increasingly intertwined, ESC appears uniquely positioned to contribute to a new generation of intelligent autonomous systems, pointing toward what may become the field's fourth scientific revolution.
ER -