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In this paper, we aim to investigate the synchronization problem of dynamicalsystems, which can be of generic linear or Lipschitz nonlinear type,communicating over directed switching network topologies. A mild connectivityassumption on the switching topologies is imposed, which allows them to bedirected and jointly connected. We propose a novel analysis framework from bothalgebraic and geometric perspectives to justify the attractiveness of thesynchronization manifold. Specifically, it is proven that the complementaryspace of the synchronization manifold can be spanned by certain subspaces.These subspaces can be the eigenspaces of the nonzero eigenvalues of Laplacianmatrices in linear case. They can also be subspaces in which the projection ofthe nonlinear self-dynamics still retains the Lipschitz property. This allowsto project the states of the dynamical systems into these subspaces andtransform the synchronization problem under consideration equivalently into aconvergence one of the projected states in each subspace. Then, assuming thejoint connectivity condition on the communication topologies, we are able towork out a simple yet effective and unified convergence analysis for both typesof dynamical systems. More specifically, for partial-state coupled genericlinear systems, it is proven that synchronization can be reached if an extracondition, which is easy to verify in several cases, on the system dynamics issatisfied. For Lipschitz-type nonlinear systems with positive-definite innercoupling matrix, synchronization is realized if the coupling strength is strongenough to stabilize the evolution of the projected states in each subspaceunder certain conditions.

On Synchronization of Dynamical Systems over Directed Switching Topologies: An Algebraic and Geometric Perspective