Hyper-Exponential Synchronization of Directed Higher-Order Networks

This paper considers the hyper-exponential and practical prescribed-time synchronization of directed higherorder networks (DHONs) with or without external disturbances. Firstly, a time-varying regulatory function for hyper-exponential stability is proposed via a recursive formula. In virtue of it, we establish several new lemmas and theorems, extending the existing hyper-exponential stability theory and connecting it with the practical prescribed-time synchronization. Through rigorous mathematic proofs, the sufficient conditions for achieving the practical prescribed-time synchronization hyper-exponentially are derived. Next, in the basis of hyper-exponential stability, we raise a class of generalized multiphase functions for multiphase synchronization, which encompasses the hyper-exponential, hyper-polynomial and hyper-logarithmic stability. Then, we apply the stability theory on the synchronization and control problems for DHONs. By defining two new transformation vectors, we convert the general hypergraph-based DHON model into a multiweighted network. With the help of the rearranging variables' order technique, all coupling matrices are combined, the corresponding normalized left eigenvectors of the sum matrices are determined for constructing the Lyapunov function, resulting in synchronization criteria. Finally, four numerical simulations are presented to validate the theoretical findings.