Fixed-Time Synchronization of Second-Order Kuramoto Oscillators With Energy Disparity Bound

This brief addresses phase locking and frequency synchronization in undirected networks of second-order Kuramoto oscillators with bounded intrinsic-frequency heterogeneity. Existing fixed-time designs for oscillator networks often rely on nonsmooth feedback and rarely quantify how regulation effort is distributed across nodes. We propose a smooth fully distributed controller that reinforces coupling and applies a two-regime fixed-time injection with hyperbolic-tangent smoothing, enabling practical fixed-time convergence without discontinuous switching. A sharp pseudoinverse-based tuning condition is derived to guarantee entry into and invariance of a phase-cohesive region, which yields uniform sector bounds for the sinusoidal coupling along the closed-loop trajectory. Within this cohesive regime, we establish a uniform practical fixed-time settling-time bound independent of initial conditions and characterize the tunable residual accuracy induced by smoothing and heterogeneity. We further derive an explicit tunable upper bound on the disparity of accumulated control energy across nodes, linking fast synchronization to balanced actuation burden. Numerical examples demonstrate multi-trajectory synchronization, cohesiveness preservation, and the resulting energy-disparity behavior.