Non-Local Extremum Seeking Based on the Divergence Theorem

Abstract: We propose a new design strategy for extremum seeking control for a multi-dimensional single-integrator system in the presence of local extrema. The proposed method employs suitably designed sinusoidal dither signals, which force the single-integrator to a spherical motion. Over time, this spherical motion gives approximate access to an integral of the objective function over a sphere. Using the divergence theorem, we identify the integral over the sphere as the gradient of an integral over the enclosed ball. This integral over the ball defines a locally averaged objective function. The proposed extremum seeking method drives the system state into the gradient direction of the averaged objective function. Such a local average of the objective function can eliminate undesired local extrema and is therefore beneficial for global optimization. Under the assumption that the averaged objective function has no undesired critical points, we prove practical asymptotic stability of the closed-loop system. Our theoretical analysis takes sufficiently small $L_\infty$-measurement errors of the objective function into account.

• The paper introduces a globally optimized extremum-seeking algorithm based on the divergence theorem by averaging spatially distributed sources to avoid local maxima.
• Key results include formula derivation via the divergence theorem to map integrand evaluations onto surface integrals, yielding an explicit gradient computation without directly calculating gradients or states. Stability analysis shows convergence of the algorithm to localized objectives, defined by ball averages with bias.
• Simulations in two,three and four dimensions demonstrate faster convergence to optimized averages than known gradient flows but pose open challenges such as handling time-variant object shapes and fine-tuning objective radius.

Problem setting and motivation

The paper addresses extremum seeking for the single-integrator system $\dot{x} = u$ in $\mathbb{R}^n$, $n \geq 2$, where the state $x$ is unmeasurable and the only feedback signal is a real-time measurement $\hat{y} = J(x) + d(t)$ of a smooth but analytically unknown objective function $J\colon\mathbb{R}^n \to \mathbb{R}$, corrupted by an $L_\infty$ disturbance $d$. Conventional dither-based extremum seeking steers the state approximately along $\nabla J$, which guarantees only convergence to the nearest local extremum. The motivating example is source seeking in fields with spatially distributed sources: the authors exhibit a radially symmetric $J$ whose gradient flow is trapped on a sphere of local maximizers of radius approximately 2.55, while the unique global maximizer sits at the origin. The stated goal is therefore global optimization via a mechanism that does not follow $\mathbb{R}^n$0.

Averaged objective and the divergence theorem

The central construction is a locally averaged objective

$\mathbb{R}^n$1

where $\mathbb{R}^n$2 is the closed unit ball and $\mathbb{R}^n$3 is a user-chosen averaging radius. Local averaging "washes out" local extrema whose width is smaller than the diameter $\mathbb{R}^n$4: as $\mathbb{R}^n$5, $\mathbb{R}^n$6 converges locally uniformly to $\mathbb{R}^n$7, whereas sufficiently large $\mathbb{R}^n$8 can eliminate all critical points except the global maximizer. A componentwise application of the divergence theorem converts the ball integral into a surface integral,

$\mathbb{R}^n$9

so that the gradient of the averaged objective depends only on point evaluations of $n \geq 2$0 — precisely what the output measurements provide. This distinguishes the approach from earlier numerical schemes based on the same identity (e.g., zeroth-order methods à la Gupal, Mayne, Flaxman), which require known $n \geq 2$1 and $n \geq 2$2 to evaluate the spherical integral directly. It also gives a mathematically precise version of the "averaging smooths out local extrema" intuition conjectured in prior perturbation-based work.

Dither design and closed-loop system

The surface integral is evaluated without knowledge of $n \geq 2$3 or $n \geq 2$4 by forcing the integrator along a nearly space-filling curve on the sphere. Using hyperspherical coordinates $n \geq 2$5 with Gram determinant weight $n \geq 2$6, the authors define angle trajectories with geometrically spaced frequencies, $n \geq 2$7, and set $n \geq 2$8. For increasing $n \geq 2$9, the curve $x$0 becomes dense in $x$1 over one period. The output feedback law combines two periodic dithers, $x$2 and $x$3:

$x$4

with $x$5 the state of a high-pass filter $x$6 (removable, but performance-enhancing). After the coordinate change $x$7, the $x$8-subsystem integrates exactly the time-parametrized integrand of the spherical integral, up to the scalar gain $x$9 and the weight $\hat{y} = J(x) + d(t)$0. In dimension $\hat{y} = J(x) + d(t)$1 the scheme coincides with a known method from the literature; the novelty lies in the multi-dimensional extension and in the interpretation via $\hat{y} = J(x) + d(t)$2, which prior analyses restricted to purely quadratic objectives did not provide.

Main stability result

The analysis proceeds in two approximation steps. First, standard Lie-bracket averaging shows that for large dither frequency $\hat{y} = J(x) + d(t)$3 the transformed closed loop tracks the averaged system with vector field $\hat{y} = J(x) + d(t)$4 plus a disturbance term $\hat{y} = J(x) + d(t)$5. Second, a quantitative equidistribution argument — based on a van der Corput-type sequence $\hat{y} = J(x) + d(t)$6 built from sawtooth waves at frequencies $\hat{y} = J(x) + d(t)$7 and controlled by the modulus of continuity of the integrand — shows that $\hat{y} = J(x) + d(t)$8 converges, as $\hat{y} = J(x) + d(t)$9, to $J\colon\mathbb{R}^n \to \mathbb{R}$0, which by the divergence-theorem identity equals $J\colon\mathbb{R}^n \to \mathbb{R}$1 with $J\colon\mathbb{R}^n \to \mathbb{R}$2. The filter contribution vanishes in the same limit.

Combining these steps yields a finite-time trajectory approximation result (Proposition 1): on compact sets, for any horizon $J\colon\mathbb{R}^n \to \mathbb{R}$3 and tolerance $J\colon\mathbb{R}^n \to \mathbb{R}$4, sufficiently large $J\colon\mathbb{R}^n \to \mathbb{R}$5 and $J\colon\mathbb{R}^n \to \mathbb{R}$6 make the closed-loop solution stay within $J\colon\mathbb{R}^n \to \mathbb{R}$7 of the gradient system $J\colon\mathbb{R}^n \to \mathbb{R}$8, uniformly over disturbances with $J\colon\mathbb{R}^n \to \mathbb{R}$9. Under Assumption 1 — compactness of a superlevel set of $L_\infty$0 and nonvanishing gradient on an annular region between two levels — this gradient system satisfies an ISS-type bound (Proposition 2), inherited from Sontag-style arguments. The main theorem then states practical asymptotic stability: for sufficiently large $L_\infty$1 and $L_\infty$2, the closed-loop state enters and remains in a neighborhood of the high superlevel set of $L_\infty$3, with a bound of the form

$L_\infty$4

The guarantee is thus practical (up to an arbitrarily small offset $L_\infty$5) and ISS-like with respect to measurement noise, but it concerns maximization of $L_\infty$6, not of $L_\infty$7 itself.

Numerical evidence

Three simulations illustrate both the capability and the boundary of the method. In dimension $L_\infty$8, for $L_\infty$9, the averaging radius $d$0 leaves residual local maxima and the closed loop converges to a spurious maximizer near $d$1, whereas $d$2 — chosen so that $d$3 exceeds the perturbation period $d$4 — yields convergence to the origin. Notably, the origin is not the global maximizer of the original $d$5; the method optimizes the averaged objective. In dimension $d$6 with the radially symmetric test function, $d$7 leads to convergence onto the sphere of local maximizers (radius ≈ 2.67), while $d$8 removes all undesired critical points of $d$9 and drives the state to the origin. A four-dimensional example with a flat maximum, $\nabla J$0, shows that averaging also reshapes the landscape to accelerate convergence: the averaged state converges noticeably faster than the gradient flow of $\nabla J$1, though the physical state $\nabla J$2 merely performs a spherical motion of radius $\nabla J$3 around the optimizer — an inherent consequence of the persistent dither.

Limitations and open questions

The paper is explicit about several caveats. There is no general guarantee that $\nabla J$4 has fewer local extrema than $\nabla J$5; verifying Assumption 1 requires knowledge of $\nabla J$6 that extremum seeking normally does not assume, and selecting $\nabla J$7 demands a priori information about the depth and width of undesired local extrema. For large $\nabla J$8, the maximizers of $\nabla J$9 can be displaced substantially from those of $J$0, so convergence to a global maximizer of $J$1 requires additional assumptions on the smallness and placement of the relevant superlevel set — assumptions that fail in the first numerical example. The stability result is practical rather than asymptotic, holds under a nonvanishing-gradient condition, and its constants depend on compact sets and disturbance bounds. The authors identify as open extensions the use of a time-varying or adaptive radius $J$2, while noting that any such design will still require some a priori knowledge of the unknown objective's shape.

Conclusion

The paper contributes a dither-design principle for multi-dimensional extremum seeking in which sinusoidal excitations confined to a sphere, combined with a Gram-determinant weighting and a high-frequency, high-density limit, convert real-time output measurements into an ascent direction for a ball-averaged objective. The divergence theorem supplies the exact link between the spherical measurement integral and the gradient of the average, and the analysis delivers a quantitative practical-ISS guarantee robust to bounded measurement errors. The method's effectiveness for global optimization is conditional on the averaging radius eliminating the relevant critical points of $J$3, a property that cannot be certified without knowledge of $J$4; characterizing when such radii exist, and designing adaptive radii, remain open problems raised by this work.