  We introduce a class of distributed nonlinear control systems, termed as the
flow-tracker dynamics, which capture phenomena where the average state is
controlled by the average control input, with no individual agent has direct
access to this average. The agents update their estimates of the average
through a nonlinear observer. We prove that utilizing a proper gradient
feedback for any distributed control system that satisfies these conditions
will lead to a solution of the corresponding distributed optimization problem.
We show that many of the existing algorithms for solving distributed
optimization are instances of this dynamics and hence, their convergence
properties can follow from its properties. In this sense, the proposed method
establishes a unified framework for distributed optimization in
continuous-time. Moreover, this formulation allows us to introduce a suit of
new continuous-time distributed optimization algorithms by readily extending
the graph-theoretic conditions under which such dynamics are convergent.
