Subspace Decomposition for Graphon LQR: Applications to VLSNs of Harmonic Oscillators
Bibliographic record
Abstract
Graphon control has been proposed and developed in [1]-[3] to approximately solve control problems for very large-scale networks of linear dynamical systems based on graphon limits. This article provides a solution method based on invariant subspace decompositions for a class of graphon linear quadratic regulation (LQR) problems, where the local dynamics share homogeneous parameters but the graphon couplings may be heterogeneous among the coupled agents. Graphon couplings in this article appear in states, controls, and costs, and these couplings may be represented by different graphons. By exploiting a common invariant subspace of the couplings, the original problem is decomposed into a network coupled LQR problem of finite dimension and a decoupled infinite dimensional LQR problem. A centralized optimal control solution, and a nodal collaborative optimal control solution, where each agent computes its part of the optimal solution locally, are established. The application of these solutions to finite network LQR problems may be via 1) the graphon control methodology [3], or 2) the representation of finite LQR problems as special cases of graphon LQR problems. The complexity of these solutions involves solving one nd×nd dimensional Riccati equation and one n×n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the nontrivial common invariant subspace of the coupling operators, whereas a direct approach involves solving an nN ×nN dimensional Riccati equation, where N is the size of the network. For situations where the graphon couplings do not admit exact low-rank representations, approximate control is developed based on low-rank approximations. Finally, an application to the regulation of harmonic oscillators coupled over large networks with uncertainties is demonstrated.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".