It is undecidable whether the growth rate of a given bilinear system is 1

TL;DR

This paper proves that it is undecidable to determine whether the growth rate of a general bilinear system equals 1, highlighting fundamental computational limits in analyzing such systems.

Contribution

It establishes the undecidability of the growth rate problem for bilinear systems and links it to the joint spectral radius computation, extending previous positive results.

Findings

No algorithm can decide if the growth rate of a bilinear system is 1.

No polynomial-time algorithm can approximate the growth rate within a relative error.

Results hold even for systems with nonnegative rational coefficients.

Abstract

We show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of $(B,v)$). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy $\varepsilon$ in time polynomial in the size of the system and of $\varepsilon$. Our two results hold even if all the coefficients are nonnegative rationals.