Optimal Time-Invariant Formation Tracking for a Second-Order Multi-Agent System

Given a multi-agent linear system, we formalize and solve a trajectory optimization problem that encapsulates trajectory tracking, distance-based formation control and input energy minimization. To this end, a numerical projection operator Newton's method is developed to find a solution by the minimization of a cost functional able to capture all these different tasks. To stabilize the formation, a particular potential function has been designed, allowing to obtain specified geometrical configurations while the barycenter position and velocity of the system follows a desired trajectory.