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 Nnu = 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, suggesting intractability in the case of two or more parameters.
Submitted, 2021. 31 pages.
© 2021 Christel Baier, Florian Funke, Simon Jantsch, Engel
Lefaucheux, Florian Luca, Joël Ouaknine, David Purser,
Markus A. Whiteland, and James Worrell.