On the Isospectral Nature of Minimum-Shear Covariance Control

TL;DR

This paper investigates the isospectral properties of minimum-shear covariance control, revealing that the evolution preserves eigenvalues and is linked to Lax isospectral flows, offering insights into control dynamics.

Contribution

It introduces an alternative formalism for reducing shear in bilinear gradient flows, highlighting the isospectral nature of the evolution and its relation to Lax flows.

Findings

The evolution preserves eigenvalues (isospectral) during control.

The formalism minimizes the eigenvalue range to reduce shear.

The control dynamics inherit properties from Lax isospectral flows.

Abstract

We revisit Brockett's attention in the context of bilinear gradient flow of an ensemble, and explore an alternative formalism that aims to reduce shear by minimizing the conditioning number of the dynamics; equivalently, we minimize the range of the eigenvalues of the dynamics. Remarkably, the evolution is isospectral, and this property is inherited by the coupled nonlinear dynamics of the control problem from a Lax isospectral flow.