  This research investigates the intricate domain of synchronization problem
among multiple agents operating within a dynamic fast switching network
topology. We concentrate on cluster synchronization within coupled linear
system under pinning control, providing both necessary and sufficient
conditions. As a pivotal aspect, this paper aim to president the weakest
possible conditions to make the coupled linear system realize cluster
synchronization exponentially. Within the context of fast switching framework,
we initially examine the necessary conditions, commencing with the
transformation of the consensus problem into a stability problem, introducing a
new variable to make the coupled system achieve cluster synchronization if the
system is controllable; communication topology switching fast enough and the
coupling strength should be sufficiently robust. Then, by using the Lyapunov
theorem, we also present that the state matrix controllable is necessary for
cluster synchronization. Furthermore, this paper culminating in the
incorporation of contraction theory and an invariant manifold, demonstrating
that the switching topology has an average is imperative for achieving cluster
synchronization. Finally, we introduce three simulations to validate the
efficacy of the proposed approach.
