Rigid systems of second-order linear differential equations

TL;DR

This paper characterizes rigid systems of second-order linear differential equations, establishing conditions for their existence and describing their structure, which has implications for understanding system stability under perturbations.

Contribution

It provides a necessary and sufficient condition for the existence of rigid systems and characterizes all such systems in the context of second-order linear differential equations.

Findings

Rigid systems exist if and only if m<n(1+√5)/2.

The paper describes all rigid systems explicitly.

It links rigidity to perturbation stability in differential systems.

Abstract

We say that a system of differential equations d^2x(t)/dt^2=Adx(t)/dt+Bx(t)+Cu(t), in which A and B are m-by-m complex matrices and C is an m-by-n complex matrix, is rigid if it can be reduced by substitutions x(t)=Sy(t), u(t)=Udy(t)/dt+Vy(t)+Pv(t) with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A,B,C. We prove that there exists a rigid system if and only if m<n(1+square_root{5})/2, and describe all rigid systems.