Research on Adaptive Inertial Control in Synchronization Systems: Based on Variational Optimization Methods and Their Applications in the Stability of Complex Networks

Abstract: Aiming at the core problem that it is difficult for a fixed inertia coefficient to balance transient disturbance suppression and long-term stability in complex network synchronization systems, an adaptive inertia control strategy based on variational optimization is proposed. Taking the Kuramoto model with inertia as the research carrier, the analytical expression of the time-varying inertia coefficient M(t) is strictly derived by the functional variational method, and a hierarchical control structure of "benchmark inertia + disturbance feedback" is constructed to achieve the organic unity of minimizing the vulnerability performance function H(T) and stability constraints. A multimodal decoupling control strategy based on Laplacian eigenvector projection is designed to enhance the feedback strength of the dominant mode by eigenvalue weighting, improving the control accuracy and dynamic response speed. Simulation verification is carried out in complex network systems, and the control performance of regular networks (RG), random networks (ER), small-world networks (SW), scale-free networks (SF) and spider webs (SP) under three typical disturbances of pulses, monotonic decays and oscillatory decays is systematically analyzed. The results show that the proposed strategy reduces H(T) of the five networks by 19%-25%, shortens the relaxation time by 15%-24%, and the real parts of all system eigenvalues are less than -0.25s^-1 , meeting the asymptotic stability criterion. This study provides a new theoretical framework and engineering implementation scheme for the stability control of complex network synchronization systems, which can be widely applied to fields such as power grids, communication networks, and neural networks.