Periodic sliding motions

A general geometric characterization is given for the global existence of sliding regimes, on compact manifolds, in nonlinear variable structure feedback systems. The characterization involves a set-theoretic inclusion condition to be satisfied by the control-dependent flow map acting on the compact region contained by the sliding manifold. A sign condition is derived on the volume integral of the divergence of the generating controlled vector field. The condition is a necessary, but not sufficient, condition for the existence of a sliding regime. The manifold invariance conditions, or ideal sliding conditions, are characterized in terms of volume-preserving evolution of the flow map associated with the sliding dynamics. An application of the general results to periodic sliding motions in R² was illustrated using some simple examples