Linear dynamical systems with continuous weight functions

TL;DR

This paper explores methods to compute various weight-based measures in linear dynamical systems with continuous and polynomial weights, including o-minimal functions, addressing ergodic properties and energy constraints.

Contribution

It introduces algorithms for calculating mean, total, and discounted weights in LDSs with continuous and polynomial weights, including o-minimal functions, and analyzes energy constraint satisfaction.

Findings

Established ergodic properties of o-minimal weight functions.

Provided algorithms for mean payoff and accumulated weight computations.

Analyzed energy constraint satisfaction in LDSs.

Abstract

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.