Nonintegrable discrete-time driftless control systems: Geometric phases beyond the area rule

Abstract

In a continuous-time nonlinear driftless control system, a geometric phase is a consequence of nonintegrability of the vector fields, and it describes how cyclic trajectories in shape space induce non-periodic motion in phase space, according to an area rule. The aim of this paper is to shown that geometric phases exist also for discrete-time driftless nonlinear control systems, but that unlike their continuous-time counterpart, they need not obey any area rule, i.e., even zero-area cycles in shape space can lead to nontrivial geometric phases. When the discrete-time system is obtained through Euler discretization of a continuous-time system, it is shown that the zero-area geometric phase corresponds to the gap between the Euler discretization and an exact discretization of the continuous-time system.