Cholesky factorisation, and intrinsically sparse linear quadratic regulation

TL;DR

This paper classifies a family of shift operator matrices that can be efficiently factorized using Cholesky, revealing intrinsic sparsity in control laws for transportation problems with tree structures, enabling distributed control.

Contribution

It introduces a classification of matrices suitable for Cholesky factorization in LQR problems, uncovering hidden sparsity and enabling distributed control strategies.

Findings

Identifies a class of shift matrices with tractable Cholesky factorization.

Reveals intrinsic sparsity in control laws for tree-structured transportation problems.

Shows that optimal control can be implemented in a distributed manner.

Abstract

We classify a family of matrices of shift operators that can be factorised in a computationally tractable manner with the Cholesky algorithm. Such matrices arise in the linear quadratic regulator problem, and related areas. We use the factorisation to uncover intrinsic sparsity properties in the control laws for transportation problems with an underlying tree structure. This reveals that the optimal control can be applied in a distributed manner that is obscured by standard solution methods.