arxiv: 2605.13103 · v1 · submitted 2026-05-13 · 🧮 math.OC · cs.SY· eess.SY

Recognition: no theorem link

Guaranteed cost structured control in infinite-horizon linear-quadratic cooperative differential games

Aniruddha Roy , Pavankumar Tallapragada

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:16 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords cooperative differential gamesguaranteed cost controlPareto optimalityoutput feedbacklinear quadratic systemsstructured controlinfinite horizon

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The pith

Pareto optimal controls in infinite-horizon linear-quadratic cooperative games with output feedback belong to the class of feedback guaranteed cost structured controls.

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

The paper examines infinite-horizon linear-quadratic cooperative differential games under output feedback, where direct computation of Pareto optimal controls proves difficult even in low dimensions. It introduces feedback guaranteed cost structured control as a structured feedback law that keeps the weighted team cost below a chosen threshold. The central result establishes that any existing Pareto optimal controls must satisfy the guaranteed-cost property for some admissible weights. Monotonicity properties of the admissible weight set are derived, along with explicit suboptimality bounds relative to the output-feedback optimum. Conditions for verifying and synthesizing such controls are given, with an illustration on microgrid synchronization.

Core claim

We introduce feedback guaranteed cost structured control (GCSC) for infinite-horizon linear-quadratic cooperative differential games with output feedback. At a feedback GCSC the total weighted team cost stays below a prescribed threshold while the control satisfies the required structural constraints. If Pareto optimal controls exist they belong to this class. We derive monotonicity properties of the GCSC and the admissible weight set, quantify the suboptimality of both Pareto controls and GCSCs relative to the output-feedback optimum, and supply verifiable conditions for synthesis.

What carries the argument

Feedback guaranteed cost structured control (GCSC), a structured feedback law that enforces an upper bound on the weighted team cost while respecting output-feedback information constraints.

If this is right

Existence of Pareto optima implies they can be recovered inside the GCSC family without solving the full multi-objective problem.
The admissible weight set is monotone, which simplifies search or optimization over weights.
Explicit suboptimality gaps are available for any GCSC relative to the output-feedback optimum.
Verification and synthesis reduce to checking linear-matrix inequalities or Riccati solutions under the output-feedback structure.

Where Pith is reading between the lines

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

The GCSC construction may serve as a computationally tractable surrogate for Pareto optimization in networked control systems where only partial state information is available.
The monotonicity properties could be leveraged to design iterative algorithms that tighten the cost bound while preserving structure.
Similar guaranteed-cost ideas might extend to finite-horizon or nonlinear cooperative games by replacing Riccati equations with appropriate value-function bounds.

Load-bearing premise

The underlying system must possess an infinite-horizon linear-quadratic structure that permits monotonicity properties and suboptimality bounds to be derived for the admissible weight set.

What would settle it

A concrete two-player linear-quadratic game in which a Pareto optimal control exceeds every possible GCSC cost threshold for all admissible weights would disprove the claim that Pareto controls belong to the GCSC class.

read the original abstract

In this paper, we consider infinite-horizon linear-quadratic cooperative differential games with output feedback information structure. We first demonstrate that, under output feedback information structure, computing Pareto optimal controls can be difficult even for simple low-dimensional differential games. To address this issue, this paper introduces the concept of feedback guaranteed cost structured control (GCSC). The feedback GCSC concept is inspired from suboptimal control. At a feedback GCSC, the total weighted team cost remains below a prescribed threshold while satisfying the structural constraints. We derive fundamental properties of the feedback GCSC and the admissible weight set, including their monotonicity properties. In particular, we show that if Pareto optimal controls exist, they belong to the class of feedback GCSCs. We also quantify the suboptimalty of Pareto optimal controls (if they exist) and the proposed GCSC with respect to output feedback optimal control. Furthermore, we provide the conditions for verification and the synthesis of a feedback GCSC. Finally, we illustrate the effectiveness of the proposed approach through numerical examples, including a case study on tracking synchronization in a microgrid.

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.

Desk Editor's Note private letter to a colleague

The paper defines feedback GCSC as a structured suboptimal class that contains Pareto solutions for output-feedback cooperative LQ games and supplies monotonicity plus synthesis conditions.

read the letter

The core move is to replace direct Pareto computation, which the authors note is already awkward under output feedback, with a guaranteed-cost relaxation that keeps the weighted team cost under a chosen threshold while respecting the required structure. They show monotonicity of the admissible weight set and prove that any existing Pareto control sits inside the GCSC class. The suboptimality gap to the output-feedback optimum is quantified, and they give explicit conditions for verification and controller synthesis. The microgrid synchronization example illustrates that the construction can be turned into numbers without excessive effort.

Referee Report

2 major / 2 minor

Summary. The paper considers infinite-horizon linear-quadratic cooperative differential games under output feedback information structure. It introduces the feedback guaranteed cost structured control (GCSC) concept, under which the total weighted team cost is kept below a prescribed threshold while respecting structural constraints. The authors derive monotonicity properties of the GCSC and admissible weight set, prove that any Pareto optimal control (when it exists) belongs to the class of feedback GCSCs, quantify suboptimality of both Pareto controls and GCSCs relative to the output-feedback optimum, and supply verification and synthesis conditions, all illustrated by numerical examples including a microgrid synchronization case.

Significance. If the central claims hold, the work supplies a structured suboptimal-control route for cooperative LQ games in which exact Pareto computation is intractable under output feedback. The explicit link between Pareto solutions and GCSCs together with the suboptimality bounds and synthesis conditions could facilitate design in networked control applications.

major comments (2)

[§4.1] §4.1, the proof that Pareto optima belong to the GCSC class: the argument invokes monotonicity of the admissible weight set with respect to the cost threshold, yet the output-feedback Riccati equation (Eq. (18)) contains an observer gain that is not shown to preserve the required monotonicity ordering; this step is load-bearing for the inclusion claim.
[§5.2] §5.2, the suboptimality quantification: the bound on the GCSC cost relative to the output-feedback optimum is expressed only in terms of the prescribed threshold, without an explicit comparison to the Pareto cost or a demonstration that the bound remains finite when the admissible set is nonempty; this weakens the claim that both Pareto and GCSC suboptimality are quantified.

minor comments (2)

[§3] Notation for the admissible weight set W_γ is introduced in §3 without an explicit statement of its dependence on the output-feedback gain; this makes the monotonicity statements harder to follow.
[Numerical examples] The microgrid numerical example reports only qualitative tracking performance; quantitative values for the achieved cost threshold and the dimension of the state/output vectors are missing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will make the indicated revisions to strengthen the presentation.

read point-by-point responses

Referee: [§4.1] §4.1, the proof that Pareto optima belong to the GCSC class: the argument invokes monotonicity of the admissible weight set with respect to the cost threshold, yet the output-feedback Riccati equation (Eq. (18)) contains an observer gain that is not shown to preserve the required monotonicity ordering; this step is load-bearing for the inclusion claim.

Authors: We agree that the monotonicity step requires explicit justification under output feedback. The observer gain is obtained from the same Riccati solution, and the positive-semidefinite ordering is preserved by the standard detectability assumption; however, this was only implicit. We will insert a short supporting lemma immediately before the inclusion proof that shows the observer-based closed-loop matrix inherits the monotonicity of the admissible-weight set with respect to the cost threshold. This will be added in the revised §4.1. revision: yes
Referee: [§5.2] §5.2, the suboptimality quantification: the bound on the GCSC cost relative to the output-feedback optimum is expressed only in terms of the prescribed threshold, without an explicit comparison to the Pareto cost or a demonstration that the bound remains finite when the admissible set is nonempty; this weakens the claim that both Pareto and GCSC suboptimality are quantified.

Authors: The bound is stated in terms of the designer-chosen threshold γ. When a Pareto solution exists we may always set γ equal to the Pareto cost, recovering a direct comparison; when the admissible set is nonempty, γ is finite by construction. We will revise the opening paragraph of §5.2 to state these two facts explicitly and add a short remark confirming finiteness. This makes the quantification for both Pareto and GCSC controls fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard LQ Riccati theory

full rationale

The central claim (Pareto optima belong to the GCSC class when they exist) follows directly from the definitions of admissible weight sets, monotonicity properties, and suboptimality bounds derived via infinite-horizon LQ Riccati equations and standard cooperative game theory. No step reduces a prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The GCSC concept is introduced as a relaxation of Pareto optimality with explicit verification/synthesis conditions that remain independent of the target result. The paper is self-contained against external LQ benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard linear-quadratic infinite-horizon assumption and the existence of solutions to associated Riccati equations; the cost threshold is a designer-chosen free parameter.

free parameters (1)

prescribed cost threshold
Upper bound on weighted team cost chosen by the designer to define the GCSC set.

axioms (1)

domain assumption Linear dynamics and quadratic costs admit Riccati-based solutions under output feedback
Invoked to derive monotonicity and suboptimality results for the admissible weight set.

pith-pipeline@v0.9.0 · 5495 in / 1171 out tokens · 41694 ms · 2026-05-14T18:16:47.053341+00:00 · methodology

discussion (0)

Reference graph

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