Linear dynamical systems with weight functions

Rajab AghamovRajab Aghamov, Christel BaierChristel Baier, Toghrul KarimovToghrul Karimov, Joël OuaknineJoël Ouaknine, Jakob PiribauerJakob Piribauer

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Rajab Aghamov, Christel Baier, Toghrul Karimov, Joël Ouaknine, Jakob Piribauer
Linear dynamical systems with weight functions
Nonlinear Analysis: Hybrid Systems, 60:101680, 2026

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KurzfassungAbstract
In discrete-time linear dynamical systems (LDSs), a linear map is repeatedly applied to an initial vector yielding a sequence of vectors called the orbit of the system. A weight function assigning weights to the points in the orbit can be used to model quantitative aspects, such as resource consumption, of a system modelled by an LDS. This paper addresses the problems of how to compute the mean payoff, the total accumulated weight, and the discounted accumulated weight of the orbit under continuous weight functions as well as polynomial weight functions as a special case. Additionally, weight functions that are definable in an o-minimal extension of the theory of the reals with exponentiation, which can be shown to be piecewise continuous, are considered. In particular, good ergodic properties of o-minimal weight functions, instrumental to the computation of the mean payoff, are established. Besides general LDSs, the special cases of stochastic LDSs and LDSs with bounded orbits are addressed. Finally, the problem of deciding whether an energy constraint is satisfied by the weighted orbit, i.e., whether the accumulated weight never drops below a given bound, is analysed.
Projekt:Project: CPEC, CeTI, SECAI
Forschungsgruppe:Research Group: Algebraische und logische Grundlagen der InformatikAlgebraic and Logical Foundations of Computer Science

@article{ABKOP2026,
  author    = {Rajab Aghamov and Christel Baier and Toghrul Karimov and
               Jo{\"{e}}l Ouaknine and Jakob Piribauer},
  title     = {Linear dynamical systems with weight functions},
  journal   = {Nonlinear Analysis: Hybrid Systems},
  volume    = {60},
  publisher = {Elsevier},
  year      = {2026},
  pages     = {101680},
  doi       = {10.1016/j.nahs.2026.101680}
}