  This paper investigates the distributed optimal output consensus problem of
second-order uncertain nonlinear multi-agent systems over weight-unbalanced
directed networks. Under the standard assumption that local cost functions are
strongly convex with globally Lipschitz gradients, a novel distributed dynamic
state feedback controller is developed such that the outputs of all the agents
reach the optimal solution to minimize the global cost function which is the
sum of all the local cost functions. The controller design is based on a
two-layer strategy, where a distributed optimal coordinator and a
reference-tracking controller are proposed to address the challenges arising
from unbalanced directed networks and uncertain nonlinear functions
respectively. A key feature of the proposed controller is that the nonlinear
functions containing the uncertainties and disturbances are not required to be
globally Lipschitz. Furthermore, by exploiting adaptive control technique, no
prior knowledge of the uncertainties or disturbances is required either. Two
simulation examples are finally provided to illustrate the effectiveness of the
proposed control scheme.
