Trajectory Convergence from Coordinate-wise Decrease of General Energy Functions

TL;DR

This paper proves that trajectories with coordinate-wise energy decrease converge to critical points or minima, extending previous convex quadratic results and applying to multi-agent systems with uncertainties.

Contribution

It introduces a general convergence result for trajectories with coordinate-wise energy decrease, broadening applicability beyond convex quadratic functions.

Findings

Trajectories converge to critical points or minima under the coordinate-wise energy decrease condition.

The results extend previous convex quadratic energy function convergence theorems.

Application demonstrated in multi-agent systems with uncertainties.

Abstract

We consider arbitrary trajectories subject to a coordinate-wise energy decrease: the sign of the derivative of each entry is never the same as that of the corresponding entry of the gradient of some energy function. We show that this simple condition guarantees convergence to a point, to the minimum of the energy functions, or to a set where its Hessian has very specific properties. This extends and strengthens recent results that were restricted to convex quadratic energy functions. We demonstrate the application of our result by using it to prove the convergence of a class of multi-agent systems subject to multiple uncertainties.