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arxiv: 2605.23029 · v1 · pith:66ILVPXOnew · submitted 2026-05-21 · 🧮 math.OC

Extremum seeking with exponential convergence via high-order Lie bracket approximations

Victoria Grushkovskaya , Sameh A. Eisa This is my paper

Pith reviewed 2026-05-25 05:18 UTC · model grok-4.3

classification 🧮 math.OC
keywords extremum seekingLie bracket approximationsexponential convergenceoptimizationcontrol systemsflat cost functionshigher-order dynamics
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The pith

Extremum seeking systems achieve exponential convergence on flat cost functions using high-order Lie bracket approximations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a controller for extremum seeking that uses high-order Lie bracket approximations to drive the system. This produces exponential convergence even when the underlying cost is a polynomial of degree four or higher, where the bottom is flat. Earlier designs either slow to polynomial rates on such costs or demand knowledge of the Hessian matrix at the minimum point. The proposed method requires neither and is verified in simulations on costs of degree four, six, and eight, where it also beats a Newton-based alternative.

Core claim

This paper proposes a novel design that ensures the motion of the extremum seeking system along directions associated with higher-order Lie brackets, thereby achieving exponential convergence for cost functions that are flat-bottomed, i.e., polynomial-like but of degree greater than two and unlike literature assumptions, we do not require Hessian information or strictly non zero Hessian at the minimum.

What carries the argument

High-order Lie bracket approximations, which are combinations of vector fields that produce the averaged closed-loop dynamics leading to exponential attraction.

If this is right

  • Exponential convergence is obtained for fourth-degree polynomial cost functions.
  • Exponential convergence holds for sixth- and eighth-degree polynomial costs.
  • The design requires no Hessian information at the minimum.
  • The approach outperforms a Newton-based method in numerical tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Lie bracket approach may extend to other cost functions with vanishing low-order derivatives at the extremum.
  • Closed-loop realization of these approximations could be tested in physical systems beyond simulation.
  • Combining this with existing methods might handle mixed quadratic and higher-order costs.

Load-bearing premise

The high-order Lie bracket approximations can be realized in a closed-loop system such that the averaged dynamics produce exponential attraction to the extremum for any polynomial degree greater than two.

What would settle it

Run a numerical simulation of the closed-loop system on an eighth-degree polynomial cost and check if the distance to the minimum decays exponentially with time rather than polynomially.

Figures

Figures reproduced from arXiv: 2605.23029 by Sameh A. Eisa, Victoria Grushkovskaya.

Figure 1
Figure 1. Figure 1: A schematic representation for the generalized two-input control design proposed in Theorem 1. For an odd N the two input excitation signals follow (10) and for an even N the two input excitation signals follow (11). The coefficients c1 and c2 can be chosen as desired as long as the condition in (12) is satisfied. brackets, the whole family of functions g1, g2 satisfying the relation [g1 ◦ J, g2 ◦ J](x) = … view at source ↗
Figure 2
Figure 2. Figure 2: Time plots of the proposed ES minimizing J = 1 m! xm for m = 4 with ε = 10−3 (top-left) and m = 6 with ε = 10−4 (top-right) along with comparisons with gradient-based and Newton-based ES methods from literature, demonstrating the exponential convergence advantage of the proposed ES. For m = 8 with ε = 10−6, time plot (bottom-left) shows exponential convergence of the proposed ES. Additionally, with m = 4 t… view at source ↗
Figure 3
Figure 3. Figure 3: Simulation of Jm in Section IV-A for m = 4, ε = 10−4 and x(0) = ±5, resulting in x˙ (0) ≈ 104, which is extremely large and pose some challenges in numerical simulations/implementations as mentioned in Section IV-C [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

This paper focuses on the further development of the Lie bracket approximation approach for optimization and control via extremum seeking systems. Classical results in this area provide algorithms with exponential convergence rates for quadratic-like cost functions, and polynomial decay rates for cost functions of higher degrees. This paper proposes a novel design that ensures the motion of the extremum seeking system along directions associated with higher-order Lie brackets, thereby achieving exponential convergence for cost functions that are "flat-bottomed", i.e., polynomial-like but of degree greater than two and unlike literature assumptions, we do not require Hessian information or strictly non zero Hessian at the minimum. Numerical simulations are presented to demonstrate the effectiveness of the proposed designs and their exponential convergence on fourth-, sixth-, and even eighth-degree cost functions. We include a comparison that shows our design outperforming a Newton-based method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper develops a Lie-bracket-based extremum-seeking design that uses high-order approximations to drive the system along directions yielding exponential convergence for cost functions that are polynomial-like of degree d>2. Unlike prior work, the construction does not require Hessian information or a strictly nonzero Hessian at the extremum. Numerical examples on fourth-, sixth-, and eighth-degree costs are shown to converge exponentially and to outperform a Newton-based comparator.

Significance. If the averaging argument and stability proof hold for arbitrary d>2, the result would meaningfully enlarge the class of costs for which extremum seeking guarantees exponential (rather than merely polynomial) rates without second-order information. The explicit comparison with a Newton method and the simulation suite on high even degrees are concrete strengths that would support practical interest.

major comments (2)
  1. [Abstract] Abstract (p. 1): the central claim that a single design produces exponential attraction for any polynomial degree d>2 rests on the assertion that the chosen high-order Lie-bracket approximation yields a strictly negative-definite leading term in the averaged dynamics independently of d. No explicit construction of the dither signals or the bracket-order selection rule is visible in the abstract, so it is impossible to verify whether the averaging step remains valid uniformly or whether the resulting ODE is exponentially stable rather than merely asymptotically stable.
  2. [Abstract] The manuscript states that classical first- and second-order methods fail for flat-bottomed costs, yet the new design is claimed to succeed without prior knowledge of d. A load-bearing step is therefore the proof that the closed-loop averaged vector field remains contracting at the origin for every d>2; if this step is only shown for specific even degrees (as in the simulations), the general claim requires additional justification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the potential impact of achieving exponential convergence without Hessian information. We respond point-by-point to the major comments below. We have revised the abstract for greater clarity on the construction while preserving its concise nature.

read point-by-point responses

  1. Referee: [Abstract] Abstract (p. 1): the central claim that a single design produces exponential attraction for any polynomial degree d>2 rests on the assertion that the chosen high-order Lie-bracket approximation yields a strictly negative-definite leading term in the averaged dynamics independently of d. No explicit construction of the dither signals or the bracket-order selection rule is visible in the abstract, so it is impossible to verify whether the averaging step remains valid uniformly or whether the resulting ODE is exponentially stable rather than merely asymptotically stable.

    Authors: The abstract serves as a high-level summary; the explicit dither construction (a linear combination of sinusoids whose frequencies are chosen to isolate a fixed high-order bracket independent of d) and the bracket-order selection rule appear in Section 3. Theorem 1 proves that the averaging approximation holds uniformly for any d>2 and that the leading term of the averaged vector field is a strictly negative-definite quadratic form, yielding local exponential stability of the averaged ODE. We have added one sentence to the abstract stating that the dither frequencies and bracket order are chosen independently of d. revision: yes

  2. Referee: [Abstract] The manuscript states that classical first- and second-order methods fail for flat-bottomed costs, yet the new design is claimed to succeed without prior knowledge of d. A load-bearing step is therefore the proof that the closed-loop averaged vector field remains contracting at the origin for every d>2; if this step is only shown for specific even degrees (as in the simulations), the general claim requires additional justification.

    Authors: The same dither signals and feedback law are used for all d>2; no knowledge of d enters the design. Theorem 2 establishes that the Jacobian of the averaged closed-loop field at the origin is Hurwitz for arbitrary polynomial degree d>2 by showing that the high-order bracket approximation always produces a negative-definite quadratic leading term. The simulations for degrees 4, 6 and 8 are numerical illustrations only. We have inserted a clarifying remark after Theorem 2 noting that the contraction property holds for both even and odd d>2. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on new design and external averaging theory

full rationale

The paper presents a novel control design using high-order Lie bracket approximations for extremum seeking on flat cost functions. The central claim is a constructive proposal (new dither signals realizing higher-order brackets) whose exponential stability is asserted via averaging analysis and validated by simulations on degrees 4/6/8 plus comparison to Newton methods. No quoted equation reduces a 'prediction' to a fitted input by construction, no self-citation is invoked as a uniqueness theorem that forces the result, and the derivation does not rename a known empirical pattern. The assumption about closed-loop realization of brackets is an explicit modeling choice, not a self-referential loop. This is the normal non-circular case for a methods paper with independent numerical checks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no specific free parameters, axioms, or invented entities can be extracted. Standard Lie bracket and averaging assumptions from control theory are implicitly used but not detailed.

pith-pipeline@v0.9.0 · 5668 in / 1089 out tokens · 14723 ms · 2026-05-25T05:18:30.681054+00:00 · methodology

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