An Input–Output Framework for Submanifold Stabilization

Abstract

We study submanifold stabilization problems from an input–output perspective, where plant and controller are relations on their sets of input–output signals. In contrast to the classical input–output approaches, we consider signals whose integral $p$ -fold distance to a submanifold is finite. For feedback interconnections of relations on such signals, we develop a framework to show that the distance of the output of the plant to the desired submanifold remains bounded. Within this framework, we present a small-gain theorem, a feedback theorem for conic relations, and a feedback theorem for passive relations. We connect our findings to multiplier theory and present applications to synchronization and pattern generation.