Improved Approximation Bounds for Moore-Penrose Inverses of Banded Matrices with Applications to Continuous-Time Linear Quadratic Control

TL;DR

This paper develops improved exponential approximation bounds for Moore-Penrose inverses of banded matrices, with applications to continuous-time linear quadratic control, enhancing understanding of inverse decay properties and stability in control systems.

Contribution

It introduces tighter bounds for the pseudoinverse of banded matrices, including decay estimates and singular value bounds, with novel applications to control theory.

Findings

Exponential decay of off-diagonal blocks in pseudoinverses

Improved bounds on singular values for saddle point systems

Perturbation bounds for continuous-time optimal control

Abstract

We present improved approximation bounds for the Moore-Penrose inverses of banded matrices, where the bandedness is induced by a metric on the index set. We show that the pseudoinverse of a banded matrix can be approximated by another banded matrix, and the error of approximation is exponentially small in the ratio of the bandwidth of the approximation to that of the original matrix. An intuitive corollary can be obtained: the off-diagonal blocks of the pseudoinverse decay exponentially with the distance between the node sets associated with row and column indices, on the given metric space. Our bounds are expressed in terms of the bound of singular values of the system. For saddle point systems, commonly encountered in optimization, we provide the bounds of singular values associated under standard regularity conditions. Remarkably, our bounds improve previously reported ones and allow us to establish a perturbation bound for continuous-domain optimal control problems by analyzing the asymptotic limit of their finite difference discretization, which has been challenging with previously reported bounds.