The Orbit Problem for Parametric Linear Dynamical Systems

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with four or more parameters. More precisely, consider $M$ a d-dimensional square matrix whose entries are rational functions in one or more real variables. Given initial and target vectors $u,v \in \mathbb{Q}^d$, the parametrised point-to-point reachability problem asks whether there exist values of the parameters giving rise to a concrete matrix $N \in \mathbb{R}^{d\times d}$, and a positive integer $n$, such that $N^n u = v$. We show decidability in the case where $M$ depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem problem for linear recurrence sequences, indicating intractability in the case of four or more parameters.