An important variant of the linear model is the delayed one where it is discussed

in great detail under two theoretical frameworks: a variational stability analysis

based on fixed point theory arguments and a standard Lyapunov-based analysis.

The investigation revisits scalar variation unifying the behavior of old biologically

inspired model and extends to the multi-dimensional (consensus) alternatives. We

compare the two methods and assess their applicability and the strength of the

results they provide whenever this is possible.

The obtained results are applied to a number of nonlinear consensus networks.

The first class of networks regards couplings of passive nature. The model is considered

on its delayed form and the linear theory is directly applied to provide strong

convergence results. The second class of networks is a generally nonlinear one and

the study is carried through under a number of different conditions. In additions the

non-linearity of the models in conjunction with delays, allows for new type of synchronized solutions. We prove the existence and uniqueness of non-trivial periodic

solutions and state sufficient conditions for its local stability. The chapter concludes

with a third class of nonlinear models. We introduce and study consensus networks

of neutral type. We prove the existence and uniqueness of a consensus point and

state sufficient conditions for exponential convergence to it.

The discussion continues with the study of a second order flocking network of

Cucker-Smale or Motsch-Tadmor type. Based on the derived contraction rates in the

linear framework, sufficient conditions are established for these systems' solutions to

exhibit exponentially fast asymptotic velocity. The network couplings are essentially

state-dependent and non-uniform and the model is studied in both the ordinary and the delayed version. The discussion in flocking models concludes with two noisy

networks where convergence with probability one and in the r-th square mean is

proved under certain smallness conditions.

The linear theory is, finally, applied on a classical problem in electrical power

networks. This is the economic dispatch problem (EDP) and the tools of the linear

theory are used to solve the problem in a distributed manner. Motivated by the

emerging field of Smart Grid systems and the distributed control methods that are

needed to be developed in order to t their architecture we introduce a distributed

optimization algorithm that calculates the optimal point for a network of power

generators that are needed to operate at, in order to serve a given load. In particular,

the power grid of interconnected generators and loads is to be served at an optimal

point based on the cost of power production for every single power machine. The

power grid is supervised by a set of controllers that exchange information on a

different communication network that suffers from delays. We define a consensus

based dynamic algorithm under which the controllers dynamically learn the overall

load of the network and adjust the power generator with respect to the optimal

operational point.