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Linear Dynamical Systems, both discrete and continuous, are invaluablemathematical models in a plethora of applications such the verification ofprobabilistic systems, model checking, computational biology, cyber-physicalsystems, and economics. We consider discrete Linear Recurrence Sequences andcontinuous C-finite functions, i.e. solutions to homogeneous LinearDifferential Equations. The Ultimate Positivity Problem gives the recurrencerelation and the initialisation as input and asks whether there is a step $n_0$(resp. a time $t_0$) such that the Linear Recurrence Sequence $u[n] \ge 0$ for$n > n_0$ (resp. solution to homogeneous linear differential equation $u(t) \ge0$ for $t > t_0$). There are intrinsic number-theoretic challenges to surmountin order to decide these problems, which crucially arise in engineering and thepractical sciences. In these settings, the difficult corner cases are seldomrelevant: tolerance to the inherent imprecision is especially critical. We thuscharacterise \textit{robust} instances of the Ultimate Positivity Problem,i.e.\ inputs for which the decision is locally constant. We describe the setsof Robust YES and Robust NO instances using the First Order Theory of theReals. We show, via the admission of quantifier elimination by the First OrderTheory of the Reals, that these sets are semialgebraic.

Characterising Robust Instances of Ultimate Positivity for Linear Dynamical Systems