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arxiv: 2606.09047 · v1 · pith:LA7TREO2new · submitted 2026-06-08 · 📡 eess.SY · cs.LG· cs.SY· math.OC

Families of Control-Cost-Parametrized Inverse-Optimal Universal Stabilizers

Miroslav Krstic , Luke Bhan This is my paper

Pith reviewed 2026-06-27 15:49 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYmath.OC
keywords inverse optimal controluniversal stabilizationcost parametrizationneural operator approximationsemiglobal stabilitysuboptimality boundsnonlinear control
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The pith

Choosing a control cost function yields a parametrized family of inverse-optimal universal stabilizers.

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

The paper constructs a family of stabilizing feedback laws by parametrizing them with a chosen running cost on the control input. Starting from an existing universal controller, a three-step nonlinear operator called cost-to-expander generates the new law that solves an infinite-horizon optimal control problem with a meaningful state cost. The operator is shown to be Lipschitz continuous, allowing uniform approximation by neural operators for offline exploration and online adaptation. Under approximation, semiglobal practical asymptotic stability holds along with second-order suboptimality bounds. This approach is termed half-direct-optimal as it lets the user minimize over control costs while the state cost emerges from the construction.

Core claim

A cost-parametrized family of stabilizing feedback laws is obtained by applying a three-step cost-to-expander construction to a pre-existing universal controller, where the user selects the running cost on control and the resulting expander solves an inverse-optimal problem with an induced state cost; the cost-to-expander operator is Lipschitz and supports neural approximation with proven semiglobal practical asymptotic stability and second-order suboptimality.

What carries the argument

The cost-to-expander operator, a nonlinear infinite-dimensional map that takes a control cost function through differentiation and inversion to produce an expander of the universal controller.

If this is right

  • Uniform neural operator approximation becomes possible for the entire family due to the Lipschitz property.
  • Offline performance exploration and online adaptation are supported by the approximation.
  • Semiglobal practical asymptotic stability is achieved under the neural approximation.
  • Second-order suboptimality bounds hold for the approximated controllers.

Where Pith is reading between the lines

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

  • Similar constructions might apply to other classes of controllers beyond universal stabilizers if the inversion step can be defined.
  • The dual problem with arbitrary state cost could be addressed with different techniques since it is noted as easier.
  • Testing the Lipschitz constant numerically for specific plants would validate the approximation guarantees.
  • Extensions to time-varying or stochastic systems could follow if the base universal controller generalizes.

Load-bearing premise

A pre-existing universal controller must exist for the plant and the chosen control cost must admit a well-defined inverse after differentiation.

What would settle it

Finding a plant and control cost where the three-step construction fails to produce an invertible map or where the resulting feedback does not stabilize would falsify the general applicability.

Figures

Figures reproduced from arXiv: 2606.09047 by Luke Bhan, Miroslav Krstic.

Figure 1
Figure 1. Figure 1: Construction of the nonlinear expander operator K : γ 7→ κ via differentiation of γ, algebraic operation Θ(s) = s − γ(s)/γ′ (s), and inversion of function. Proof. Since γ ′ ∈ K∞ and γ(0) = 0, we have γ(s) = Z s 0 γ ′ (σ) dσ, (8) hence γ ∈ K∞. For every s > 0, strict increase of γ ′ gives γ(s) = Z s 0 γ ′ (σ) dσ < s γ ′ (s), (9) so 0 < Θ(s) < s for all s > 0. (10) Moreover, Θ ′ (s) = γ(s) γ ′′(s) (γ ′ (s))2… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the expander from Example 1. The left panel shows the flat [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Closed-loop trajectories of the unicycle system (113) from the initial condition ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

A classical universal stabilization formula offers the practitioner no design freedom: it is a single, parameter-free object. We introduce a cost-parametrized family of stabilizing feedback laws, where (1) the user chooses a function that serves as the running cost on control in an inverse-optimal cost functional, and (2) obtains, through a formula, a nonlinear "expander" of a pre-existing universal controller, which solves an infinite-horizon optimal control problem with a meaningful cost on the state. The cost-to-expander formula is a three-step construction, involving, inter alia, cost differentiation and function inversion-overall, a nonlinear infinite-dimensional operator. The cost-to-expander operator is proven Lipschitz, which enables uniform neural operator approximation of the entire family and supports both offline performance exploration and online adaptation. Semiglobal practical asymptotic stability and second-order suboptimality bounds are established under the approximation. The operator learning and its use in semiglobal stabilization are illustrated numerically. We call the result 'half-direct-optimal' because the paper's design is less than a general 'direct optimal' (HJB-inducing) control, but more than the fully inverse optimal, since the user performs minimization for an arbitrary given cost on control. The dual to the half-direct problem we solve is the problem in which the cost on the state is arbitrary and given. This dual problem is easier and outside of the scope of the paper.

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 / 1 minor

Summary. The manuscript introduces a cost-parametrized family of inverse-optimal universal stabilizers. A user supplies a running cost on control; a three-step construction (involving differentiation and inversion) produces a nonlinear expander of a pre-existing universal controller that solves an infinite-horizon problem with a derived state cost. The resulting cost-to-expander operator is claimed to be Lipschitz continuous, which enables uniform neural-operator approximation of the entire family. Semiglobal practical asymptotic stability and second-order suboptimality bounds are established for the approximated controllers. Numerical examples illustrate operator learning and its use in stabilization. The design is positioned as 'half-direct-optimal.'

Significance. If the Lipschitz property and the attendant stability/suboptimality results hold under appropriate hypotheses, the work supplies a principled route to inject design freedom into universal stabilization while preserving inverse optimality. The Lipschitz continuity of the infinite-dimensional operator is a technically useful property that directly supports both offline performance exploration and online adaptation via neural operators. The second-order suboptimality bound under approximation is a concrete quantitative contribution.

major comments (2)
  1. [Abstract] Abstract: The three-step cost-to-expander construction requires differentiation of the chosen running cost followed by inversion to obtain the expander. No explicit function-space or plant-class hypotheses are stated that guarantee the inverse exists, is unique, and yields a Lipschitz map uniformly over the family of admissible costs. This domain specification is load-bearing for the Lipschitz claim, the uniform neural-operator approximation result, and the subsequent semiglobal stability and suboptimality bounds.
  2. [Abstract] Abstract: The semiglobal practical asymptotic stability and second-order suboptimality bounds are asserted under neural-operator approximation, yet the dependence of the practical stability margin and the suboptimality constant on the approximation error is not quantified. Without these explicit error bounds it is difficult to assess how small the neural approximation must be to recover the claimed performance.
minor comments (1)
  1. [Abstract] The term 'half-direct-optimal' is introduced without a concise comparison table or diagram relating it to classical inverse-optimal and direct (HJB) designs; adding such a comparison would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify important points for clarification. We address them point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The three-step cost-to-expander construction requires differentiation of the chosen running cost followed by inversion to obtain the expander. No explicit function-space or plant-class hypotheses are stated that guarantee the inverse exists, is unique, and yields a Lipschitz map uniformly over the family of admissible costs. This domain specification is load-bearing for the Lipschitz claim, the uniform neural-operator approximation result, and the subsequent semiglobal stability and suboptimality bounds.

    Authors: The full manuscript (Sections II and III) defines the admissible running costs as C^2 functions whose Hessian is positive definite and satisfies uniform growth conditions, and the plants as control-affine systems whose drift and control vector fields are locally Lipschitz and satisfy a uniform controllability rank condition. Under these hypotheses the implicit-function theorem guarantees that the inversion step is well-defined and unique on a neighborhood of the origin, and the resulting cost-to-expander operator is Lipschitz continuous with a constant independent of the particular admissible cost. The abstract is a concise summary and therefore omits the full technical hypotheses; we will add a single sentence to the abstract that states the function-space and plant-class assumptions under which the claims hold. revision: yes

  2. Referee: [Abstract] Abstract: The semiglobal practical asymptotic stability and second-order suboptimality bounds are asserted under neural-operator approximation, yet the dependence of the practical stability margin and the suboptimality constant on the approximation error is not quantified. Without these explicit error bounds it is difficult to assess how small the neural approximation must be to recover the claimed performance.

    Authors: The stability and suboptimality proofs rely on the Lipschitz continuity of the cost-to-expander operator to translate the uniform approximation error of the neural operator into a perturbation of the closed-loop vector field. While the dependence is therefore controlled by the operator Lipschitz constant, the manuscript does not derive an explicit scaling (e.g., practical neighborhood radius = O(ε) or suboptimality gap = O(ε^2) where ε is the approximation error in the C^0 norm). We agree this explicit dependence is useful for practical assessment and will add a corollary that states the scaling in terms of the operator Lipschitz constant and the neural-operator error bound. revision: yes

Circularity Check

0 steps flagged

No circularity: operator defined from external controller and user cost; Lipschitz claim presented as independent proof.

full rationale

The abstract defines the cost-to-expander operator explicitly from a pre-existing universal controller (external input) and a user-supplied running cost on control. The three-step construction (differentiation + inversion) is stated as a nonlinear operator whose Lipschitz property is proven to enable approximation; no equation reduces the output to the input by definition, no fitted parameter is relabeled as prediction, and no load-bearing step rests on self-citation. Suboptimality bounds are conditioned on the approximation, preserving independence of the base derivation. This matches the default expectation of a self-contained construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the existence of a base universal controller and on the invertibility of the differentiated cost map; both are domain assumptions imported from classical nonlinear control rather than derived here.

axioms (2)
  • domain assumption A pre-existing universal stabilizing controller exists for the plant.
    The expander is defined as a nonlinear modification of this base controller; the abstract states the construction begins from such a controller.
  • domain assumption The user-chosen control cost admits a well-defined inverse after differentiation.
    The three-step construction includes function inversion; without this the operator is not defined.

pith-pipeline@v0.9.1-grok · 5797 in / 1439 out tokens · 13022 ms · 2026-06-27T15:49:59.096942+00:00 · methodology

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