  This paper addresses the optimal time-invariant formation tracking problem
with the aim of providing a distributed solution for multi-agent systems with
second-order integrator dynamics. In the literature, most of the results
related to multi-agent formation tracking do not consider energy issues while
investigating distributed feedback control laws. In order to account for this
crucial design aspect, we contribute by formalizing and proposing a solution to
an optimization problem that encapsulates trajectory tracking, distance-based
formation control, and input energy minimization, through a specific and key
choice of potential functions in the optimization cost. To this end, we show
how to compute the inverse dynamics in a centralized fashion by means of the
Projector-Operator-based Newton's method for Trajectory Optimization (PRONTO)
and, more importantly, we exploit such an offline solution as a general
reference to devise a stabilizing online distributed control law. Finally,
numerical examples involving a cubic formation following a straight path in the
3D space are provided to validate the proposed control strategies.
