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 two or more parameters. More precisely, consider a d-dimensional square matrix M whose entries are algebraic functions in one or more real variables. Given initial and target vectors u, v ∈ Qd, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix N ∈ Rd×d, and a positive integer n ∈ N, such that N nu = v.
We show decidability for the case in which M depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.