# Vector Contraction Analysis for Nonlinear Dynamical Systems

## TL;DR

This paper introduces vector contraction analysis, a novel extension of contraction theory using vector-valued norms, enabling more flexible convergence analysis of nonlinear systems through comparison systems and cone ordering.

## Contribution

It develops a new vector contraction framework that relaxes strict conditions of standard contraction analysis and applies comparison principles and cone ordering for nonlinear system analysis.

## Key findings

- Vector contraction analysis generalizes standard contraction theory.
- Convergence can be established using comparison systems and vector inequalities.
- Framework is demonstrated through illustrative examples.

## Abstract

This paper derives new results for the analysis of nonlinear systems by extending contraction theory in the framework of vector distances. A new tool, vector contraction analysis utilizing a notion of the vector-valued norm which evidently induces a vector distance between any pair of trajectories of the system, offers an amenable framework as each component of vector-valued norm function satisfies fewer strict conditions as that of standard contraction analysis. Particularly, every element of vector-valued norm derivative need not be strictly negative definite for convergence of any pair of trajectories of the system. Moreover, vector-valued norm derivative satisfies a component-wise inequality employing some comparison system. In fact, the convergence analysis is performed by comparing the relative distances between any pair of the trajectories of the original nonlinear system and the comparison system. Comparison results are derived by utilizing the concepts of quasi-monotonicity property of the function and vector differential inequalities. Moreover, the results are also derived in the framework of the cone ordering instead of utilizing component-wise inequalities between vectors. In addition, the proposed framework is illustrated by examples.