Finite-time synchronization of Kuramoto-type oscillators

In this paper, we discuss finite-time phase-frequency synchronization for Kuramoto oscillators. To achieve finite-time convergence, we study modifications of the Kuramoto model: the normalized and signed gradient systems. For identical oscillators, we establish finite-time phase-frequency synchronization for both normalized and signed gradient type Kuramoto models when the initial phase diameter is smaller than ππ. In particular, for the initial phases whose diameter is smaller than π/2π/2, the synchronization time can be upper bounded in terms of the initial phase configurations and coupling topology. For non-identical oscillators, finite-time phase-frequency synchronization for normalized gradient type Kuramoto model is also obtained when the initial phase diameter is smaller than π/2π/2. An upper bound on the convergence time is given by the initial phase diameter and coupling topology. Our approach is based on a combination of nonsmooth stability analysis and algebraic graph theory.