Extremal collective behavior

Curves and natural frames can be used for describing and controlling motion in both biological and engineering contexts (e.g., pursuit and formation control). The geometry of curves and frames leads naturally to a Lie group formulation where coordinated motion is represented by interacting particles on Lie groups - specifically, SE(2) or SE(3). Here we consider a particular type of optimal control problem in which the interactions between particles arise from a cost function dependent on each particle's steering, and which penalizes steering differences between the particles (expressed via the graph Laplacian). With this choice of cost function, we are able to perform Lie-Poisson reduction. Furthermore, we are able to derive a closed-form expression (using Jacobi elliptic functions) for certain special solutions of the coupled multi-particle problem on SE(2).