Graphon control has been proposed and developed in [Gao and Caines CDC 2017, 2018, 2019, TAC 2020) to approximately solve control problems for very large-scale networks of linear dynamical systems. In this paper, linear quadratic regulation (LQR) problems for graphon dynamical systems are studied. Graphon couplings appear in states, controls and cost, and these couplings may be represented by different graphons. Based on invariant subspace decompositions, this work provides a solution method for a class of such problems where the local dynamics is homogeneous but the network couplings are heterogeneous among the coupled subsystems. By exploring 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 solution and a nodal collaborative optimal control solution are established. The complexity of these solutions involves solving one nd X nd dimensional Riccati equation and one n X n Riccati equation, where n is the dimension of each nodal agent state and d is the dimension of the (nontrivial) invariant subspace shared by the coupling operators. 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.

中文翻译：

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Graphon LQR 的子空间分解：谐波振荡器在 VLSN 中的应用

[Gao and Caines CDC 2017, 2018, 2019, TAC 2020) 中提出并开发了 Graphon 控制，以近似解决超大规模线性动力系统网络的控制问题。在本文中，研究了图形动力学系统的线性二次调节（LQR）问题。图形耦合出现在状态、控制和成本中，这些耦合可以用不同的图形表示。该工作基于不变子空间分解，为局部动力学同质但耦合子系统间网络耦合异质的一类问题提供了一种解决方法。通过探索耦合的公共不变子空间，将原问题分解为有限维网络耦合LQR问题和解耦无限维LQR问题。建立集中最优解和节点协同最优控制解。这些解决方案的复杂性涉及求解第一个 X 维 Riccati 方程和一个 n X n Riccati 方程，其中 n 是每个节点代理状态的维度，d 是耦合算子共享的（非平凡的）不变子空间的维度。对于图子耦合不允许精确低秩表示的情况，基于低秩近似开发近似控制。最后，演示了在具有不确定性的大型网络上耦合谐波振荡器的调节的应用。其中 n 是每个节点代理状态的维度，d 是耦合算子共享的（非平凡的）不变子空间的维度。对于图子耦合不允许精确低秩表示的情况，基于低秩近似开发近似控制。最后，演示了在具有不确定性的大型网络上耦合谐波振荡器的调节的应用。其中 n 是每个节点代理状态的维度，d 是耦合算子共享的（非平凡的）不变子空间的维度。对于图子耦合不允许精确低秩表示的情况，基于低秩近似开发近似控制。最后，演示了在具有不确定性的大型网络上耦合谐波振荡器的调节的应用。