# Trajectory convergence from coordinate-wise decrease of quadratic energy functions, and applications to platoons

## TL;DR

This paper establishes convergence conditions for trajectories based on coordinate-wise energy decrease, particularly for quadratic functions, and applies these results to ensure stability in platoon control systems with measurement deadzones.

## Contribution

It introduces a convergence criterion for trajectories related to quadratic energy functions and demonstrates its application to platoon control with measurement deadzones.

## Key findings

- Convergence guaranteed for quadratic positive definite energy functions.
- Extension of results to certain semi-definite quadratic functions including graph Laplacians.
- Application to platoon stability with deadzone control laws.

## Abstract

We consider trajectories where the sign of the derivative of each entry is opposite to that of the corresponding entry in the gradient of an energy function. We show that this condition guarantees convergence when the energy function is quadratic and positive definite and partly extend that result to some classes of positive semi-definite quadratic functions including those defined using a graph Laplacian. We show how this condition allows establishing the convergence of a platoon application in which it naturally appears, due to deadzones in the control laws designed to avoid instabilities caused by inconsistent measurements of the same distance by different agents.