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 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.
Proceedings of CONCUR 21, LIPIcs 203, 2021. 18 pages.
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© 2021 Christel Baier, Florian Funke, Simon Jantsch, Toghrul Karimov, Engel
Lefaucheux, Florian Luca, Joël Ouaknine, David Purser,
Markus A. Whiteland, and James Worrell.