  In this paper, we aim to investigate the synchronization problem of dynamical
systems, which can be of generic linear or Lipschitz nonlinear type,
communicating over directed switching network topologies. A mild connectivity
assumption on the switching topologies is imposed, which allows them to be
directed and jointly connected. We propose a novel analysis framework from both
algebraic and geometric perspectives to justify the attractiveness of the
synchronization manifold. Specifically, it is proven that the complementary
space of the synchronization manifold can be spanned by certain subspaces.
These subspaces can be the eigenspaces of the nonzero eigenvalues of Laplacian
matrices in linear case. They can also be subspaces in which the projection of
the nonlinear self-dynamics still retains the Lipschitz property. This allows
to project the states of the dynamical systems into these subspaces and
transform the synchronization problem under consideration equivalently into a
convergence one of the projected states in each subspace. Then, assuming the
joint connectivity condition on the communication topologies, we are able to
work out a simple yet effective and unified convergence analysis for both types
of dynamical systems. More specifically, for partial-state coupled generic
linear systems, it is proven that synchronization can be reached if an extra
condition, which is easy to verify in several cases, on the system dynamics is
satisfied. For Lipschitz-type nonlinear systems with positive-definite inner
coupling matrix, synchronization is realized if the coupling strength is strong
enough to stabilize the evolution of the projected states in each subspace
under certain conditions.
