AI 中文总结
本文针对流值状态构建霍奇理论反馈框架，以解决带局部传感器和对偶源的标量抛物恒温器问题，提出拓扑恒温器方案，结合传输网络实例验证谐波流调节与阻尼效果。
AI 中文摘要
本文针对流值状态构建了一种霍奇理论反馈框架，其研究动机来自带有局部传感器和对偶源的标量抛物恒温器问题。在连通图上，边空间可分解为割空间与谐波循环空间。理想拓扑恒温器将割分量调节至规定的无循环传输目标，并对谐波分量进行阻尼。对于通过选定物理边通道的实现，第一贝蒂数给出了精确压缩完整谐波扇区所需的最小感知与执行秩。最小秩并不意味着非局部霍奇投影器的局部实现；剩余的割-谐波块量化了溢出与强迫。对于传输网络，我们构建了理想与有限通道非线性线驱动摆闭合系统，并在角度凝聚同步平衡点处线性化完整闭环系统。在执行器固定的情况下，被动切线摆运动会保留任何预先存在的谐波线流分量。理想闭合系统用指数衰减替代该守恒定律，并附加一个稳定的谐波块，且不改变简化后的被动节点-割谱。在通道秩条件下，有限实现可重现谐波压缩，而自主谐波衰减与完整简化状态恢复则需要额外的解耦和赫尔维茨条件。以目标为中心的布雷格曼平衡为理想度量兼容控制器以及规范共址有限通道反馈产生了非线性耗散。显式theta网络、微电网、环以及合成IEEE 14节点示例说明了传输调节、循环流阻尼以及定位诱导瞬变。实际设备动态、约束以及大规模验证仍是开放问题。
英文摘要
This paper develops a Hodge-theoretic feedback framework for flow-valued states, motivated by scalar parabolic thermostat problems with localized sensors and dual sources. On a connected graph, the edge space decomposes into a cut space and a harmonic cycle space. The ideal topological thermostat regulates the cut component toward a prescribed cycle-free transfer target and damps the harmonic component. For realizations through selected physical edge channels, the first Betti number gives the minimal sensing and actuation ranks required for exact compression of the full harmonic sector. Minimal rank does not imply local realization of the nonlocal Hodge projector; the remaining cut-harmonic blocks quantify spillover and forcing. For transmission networks, we formulate ideal and finite-channel nonlinear line-actuated swing closures and linearize the complete closed systems at an angle-cohesive synchronous equilibrium. With the actuator fixed, passive tangent swing motion preserves any pre-existing harmonic line-flow component. The ideal closure replaces this conservation law by exponential decay and appends a stable harmonic block without changing the reduced passive nodal-cut spectrum. Under the channel-rank conditions, finite realizations reproduce the harmonic compression, whereas autonomous harmonic decay and full reduced-state recovery require additional decoupling and Hurwitz conditions. A target-centred Bregman balance yields nonlinear dissipation for the ideal metric-compatible controller and for canonical colocated finite-channel feedback. Explicit theta-network, microgrid, ring, and synthetic IEEE 14-bus examples illustrate transfer regulation, cycle-flow damping, and localization-induced transients. Realistic device dynamics, constraints, and large-scale validation remain open.
Comments69 pages, 7 figures, 2 tables; ancillary Python scripts included
