The Orbit Problem for Parametric Linear Dynamical Systems

9 We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. 10 We show decidability in the case of one parameter and Skolem-hardness with two or more parameters. 11 More precisely, consider a d-dimensional square matrix M whose entries are algebraic functions in 12 one or more real variables. Given initial and target vectors u, v ∈ Q, the parametric point-to-point 13 orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix 14 N ∈ Rd×d, and a positive integer n ∈ N, such that Nu = v. 15 We show decidability for the case in which M depends only upon a single parameter, and we 16 exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting 17 intractability in the case of two or more parameters. 18 2012 ACM Subject Classification Theory of computation → Logic and verification 19

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