An Amendment to "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design"

TL;DR

This paper amends a key theorem in control contraction metrics by adding a technical condition to ensure integrability of control signals, clarifying when stabilizability is guaranteed under rank conditions of the input matrix.

Contribution

It introduces an additional technical condition to the original theorem, clarifying the circumstances under which the theorem's claims hold, especially regarding input matrix rank.

Findings

The new condition ensures integrability of differential control signals.

Counterexample shows weaker conditions may not guarantee stabilizability when input rank drops.

Remaining claims and examples remain valid with the new condition.

Abstract

We provide an amendment to the first theorem of "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" by Manchester & Slotine in the form of an additional technical condition required to show integrability of differential control signals. This technical condition is shown to be satisfied under the original assumptions if the input matrix is constant rank, and also if the strong conditions for a CCM hold. However a simple counterexample shows that if the input matrix drops rank, then the weaker conditions of the original theorem may not imply stabilizability of all trajectories. The remaining claims and illustrative examples of the paper are shown to remain valid with the new condition.