Complexity of the LTI system trajectory boundedness problem

TL;DR

This paper investigates the computational complexity of determining whether a Linear Time Invariant system with rational coefficients has bounded trajectories, revealing the problem's intricate nature and the need for careful algorithmic implementation.

Contribution

It demonstrates that classical control tools can decide boundedness in polynomial time, but require sophisticated analysis and implementation.

Findings

Classical tools can decide boundedness in polynomial time

Deciding trajectory boundedness is more complex than previously thought

Careful implementation is essential for these algorithms to work effectively

Abstract

We study the algorithmic complexity of the problem of deciding whether a Linear Time Invariant dynamical system with rational coefficients has bounded trajectories. Despite its ubiquitous and elementary nature in Systems and Control, it turns out that this question is quite intricate, and, to the best of our knowledge, unsolved in the literature. We show that classical tools, such as Gaussian Elimination, the Routh--Hurwitz Criterion, and the Euclidean Algorithm for GCD of polynomials indeed allow for an algorithm that is polynomial in the bit size of the instance. However, all these tools have to be implemented with care, and in a non-standard way, which relies on an advanced analysis.