  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.
