Noether Symmetries and Decay Laws in Formation Control of Multi-agent Systems

Distance-based formation control of agents with double integrator dynamics can be seen as a stabilization system whose evolution can be described by a time-dependent Lagrangian function. In this paper, a Noether theorem for this class of systems is obtained, giving rise to an intrinsic geometric understanding of the exponential decay for the constants of the motion of the agents, in particular, linear and angular momentum. An interesting family of geometric integrators, called variational integrators, is defined by using discretizations of the Hamiltons principle of critical action. The variational integrators preserve some geometric features such as the momentum map, and the decay of the systems energy presents a good behavior. We derive variational integrators for time-dependent Lagrangian systems that can be employed in the context of distance-based formation control algorithms. In particular, we provide an accurate numerical integrator preserving the exponential decay of the constants of motion. Copyright (C) 2021 The Authors.