Prescribed-time synchronization of directed Kuramoto oscillator networks with communication delays via pinning control

Abstract The research studies the prescribed-time synchronization of directed Kuramoto oscillator networks (KONs) subject to communication delays via pinning control. Firstly, a novel lemma on prescribed-time stability is established, in which the addition of the constant term allows the temporal derivative of the selected Lyapunov function not to be strictly non-positive over the entire interval. Secondly, based on the lemma, some sufficient conditions that guarantee prescribed-time phase synchronization (PTPS) and prescribed-time frequency synchronization (PTFS) in networks containing a directed spanning tree (DST) are derived, respectively. The two proposed control schemes guarantee the PTPS and the PTFS for such networks by controlling only the root node, while significantly reducing requirements on network connectivity, coupling strength, and initial condition distribution including both phase states and frequency states. In large-scale networks, the time delay for network-wide information exchange is non-negligible. To address this, a distributed pinning strategy, developed by extending the single root node pinning strategy, guarantees synchronization within a prescribed time horizon. Finally, the efficiency of two control schemes are demonstrated through some numerical simulations.