  This paper investigates the problem of securing exponentially fast consensus
(exponential consensus for short) for identical agents with finite-dimensional
linear system dynamics over dynamic network topology. Our aim is to find the
weakest possible conditions that guarantee exponentially fast consensus using a
Lyapunov function consisting of a sum of terms of the same functional form. We
first investigate necessary conditions, starting by examining the system (both
agent and network) parameters. It is found that controllability of the linear
agents is necessary for reaching consensus. Then, to work out necessary
conditions incorporating the network topology, we construct a set of Laplacian
matrix-valued functions. The precompactness of this set of functions is shown
to be a significant generalization of existing assumptions on network topology,
including the common assumption that the edge weights are bounded piecewise
constant functions or continuous functions. With the aid of such a
precompactness assumption and restricting the Lyapunov function to one
consisting of a sum of terms of the same functional form, we prove that a joint
$(\delta, T)$-connectivity condition on the network topology is necessary for
exponential consensus. Finally, we investigate how the above two
``necessities'' work together to guarantee exponential consensus. To partially
address this problem, we define a synchronization index to characterize the
interplay between agent parameters and network topology. Based on this notion,
it is shown that by designing a proper feedback matrix and under the
precompactness assumption, exponential consensus can be reached globally and
uniformly if the joint $(\delta,T)$-connectivity and controllability conditions
are satisfied, and the synchronization index is not less than one.
