  This paper studies cooperative control of noncooperative double-integrator
multi-agent systems (MASs) with input delay on connected directed graphs in the
context of a differential graphical game (DGG). In the distributed DGG, each
agent seeks a distributed information control policy by optimizing an
individual local performance index (PI) of distributed information from its
graph neighbors. The local PI, which quadratically penalizes the agent's
deviations from cooperative behavior (e.g., the consensus here), is constructed
through the use of the graph Laplacian matrix. For DGGs for double-integrator
MASs, the existing body of literature lacks the explicit characterization of
Nash equilibrium actions and their associated state trajectories with
distributed information. To address this issue, we first convert the N-player
DGG with m communication links into m coupled optimal control problems (OCPs),
which, in turn, convert to the two-point boundary-value problem (TPBVP). We
derive the explicit solutions for the TPBV that constitute the explicit
distributed information expressions for Nash equilibrium actions and the state
trajectories associated with them for the DGG. An illustrative example verifies
the explicit solutions of local information to achieve fully distributed
consensus.
