Isospectral Steering

TL;DR

This paper investigates the controllability of a differential Lyapunov equation using isospectral rotations, focusing on modulating eigen-vectors while keeping eigenvalues fixed, with applications in reducing anisotropic deformation.

Contribution

It introduces the concept of isospectral steering for covariance control and characterizes the reachable set of covariances under eigenvalue constraints.

Findings

Characterizes the reachable set of covariances for fixed eigenvalues.

Links the theory to multilinear algebra and positive linear algebra.

Provides a framework for controlling ensembles with minimal anisotropic deformation.

Abstract

We study the controllability of the differential Lyapunov equation under isospectral rotation of a linear gradient field. Specifically, control is effected by a symmetric time-varying gain-matrix constrained to have fixed eigenvalues; that is, by exclusively modulating the eigen-vectors of the state matrix and not its eigenvalues. Motivation for this problem stems from a certain type of control objectives (minimum shear/attention) aimed to reduce anisotropic deformation when ensembles are steered by a common law--optimality necessitates constancy of eigenvalues. In the paper we introduce and motivate this type of isospectral steering, and describe the reachable set of covariances for any specified terminal time and eigenvalues of the gain matrix. The theory we develop is intimately linked to multilinear algebra as well as to positive linear algebra and the Birkoff-von Neumann theorem for doubly stochastic matrices.