On the eigenvalues of a class of matrices with displacement structure arising in optimal control

TL;DR

This paper develops a framework for analyzing eigenvalues of structured matrices arising in optimal control, enabling explicit computation and bounds through recursive and analytical methods.

Contribution

It introduces a novel recursive characterization of characteristic polynomials for matrices with displacement structure, facilitating eigenvalue analysis and bounds.

Findings

Eigenvalues can be expressed as roots of a monotone transcendental function.

Recursive formulas for characteristic polynomials are derived.

The framework allows for analytical and numerical eigenvalue bounds.

Abstract

In this work we present a framework for studying the eigenvalues of a family of matrices with a particular displacement structure. The family admits a specific decomposition as the product of an upper and a lower triangular matrices having an increasing number of real parameters in predefined positions. Similar matrices appear naturally when solving some kinds of optimal control problems. In our case, as stated by Nehari's theorem, the eigenvalues and eigenvectors fully characterize the solution. Commonly, such problems are solved by numerical means, making it difficult to obtain insight in the role that the parameters play on the solution. Our results provide a framework that enables to compute individually, under some simple assumptions, the eigenvalues of the matrices as roots of a monotone transcendental function with many desirable properties. In order to do so, we first obtain a three-term recursive characterization of the corresponding characteristic polynomials. This enables the aforementioned representation. Our framework also allows for the computation of bounds, numerical methods and even analytical characterizations with closed form solutions, whenever the problem parameters satisfy simple conditions.