On Existence of Solutions to Structured Lyapunov Inequalities

TL;DR

This paper establishes conditions based on $\\mathcal{H}$ matrices for the existence of block-diagonal solutions to Lyapunov inequalities, aiding basis pursuit algorithms for control system stability analysis.

Contribution

It introduces new existence conditions for structured Lyapunov solutions using $\\mathcal{H}$ matrices and links them to small gain theorems, facilitating initial feasible point construction.

Findings

Derived sufficient conditions for block-diagonal Lyapunov solutions.

Linked $\\mathcal{H}$ matrices to small gain theorem applications.

Provided methods to construct solutions without full Lyapunov inequality solving.

Abstract

In this paper, we derive sufficient conditions on drift matrices under which block-diagonal solutions to Lyapunov inequalities exist. The motivation for the problem comes from a recently proposed basis pursuit algorithm. In particular, this algorithm can provide approximate solutions to optimisation programmes with constraints involving Lyapunov inequalities using linear or second order cone programming. This algorithm requires an initial feasible point, which we aim to provide in this paper. Our existence conditions are based on the so-called $\mathcal{H}$ matrices. We also establish a link between $\mathcal{H}$ matrices and an application of a small gain theorem to the drift matrix. We finally show how to construct these solutions in some cases without solving the full Lyapunov inequality.