Counting critical formations on the circle: algebraic–geometric and Morse-theoretic bounds

Abstract

In this work, we analyze the critical points of potential functions on the torus arising in decentralized formation shape control of multiagent systems on the unit circle. A rotation symmetry of the potential function allows to reduce the analysis to a reduced function on a lower-dimensional torus. We show that these reduced potential functions are generically Morse functions. Lower bounds on the number of critical points of the reduced potential are obtained by Morse-theoretic arguments. For trees, precise information on the number of critical points is obtained. We use algebraic–geometric methods to determine upper bounds on the number of critical points of the reduced potentials. Our approach is applicable both to standard formation control potential functions and potential functions arising from the Kuramoto model of coupled oscillators with identical natural frequencies.