On the Isospectral Nature of Minimum-Shear Covariance Control
This work addresses a theoretical control problem in dynamical systems, but appears incremental as it revisits and extends existing formalism.
The paper tackles the problem of reducing shear in bilinear gradient flow by minimizing the conditioning number of the dynamics, which is equivalent to minimizing the eigenvalue range, and finds that the evolution is isospectral, inheriting this property from a Lax isospectral flow.
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.