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Max-plus steady states in discrete event dynamic systems with inexact data

Abstract

Max-plus algebra is defined as the set of all real numbers with two binary operations (maximum and addition). This combination of the operations forms a very applicable tool for the investigation of systems working in discrete steps (discrete event dynamic systems). The search for the steady states in such systems leads to the study of the eigenvectors of the production matrix in the corresponding max-plus algebra. A vector x is said to be an eigenvector of a square matrix A if Ax = λx for some \(\lambda \in {\mathbb {R}}\). In real systems, the input values are usually taken to be in some interval. This paper investigates the properties of eigenspaces for vectors with interval (inexact) coefficients. We suppose that an interval vector X can be split into two subsets according to a forall–exists quantification of its interval entries, i.e., X = XX. If for any vector of X there is at least one vector of X such that their vector maximum is an eigenvector of A, then X is said to be a λ AE-eigenvector. Analogously, if there is at least one vector of X such that for any vector of X their vector maximum is an eigenvector of A, then X is said to be a λ EA-eigenvector. The properties of such eigenvectors are studied and their characterizations by equivalent conditions are presented. Polynomial and pseudopolynomial algorithms for checking some types of λ EA/λ AE-eigenvectors are suggested.

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References

  • Allamigeon X, Legay A, Fahrenberg U, Katz R, Gaubert S (2014) Tropical Fourier–Motzkin elimination, with an application to real-time verification. Internat J Algebra and Comput 24(5):569–607

    MathSciNet  Article  Google Scholar 

  • Butkovič P (2010) Max-linear systems: Theory and Algorithms. Springer, London

    Book  Google Scholar 

  • Cechlárová K (2001) Solutions of Interval Linear Systems in (max, +)-algebra. In: Proceedings of the 6th international symposium on operational research preddvor, Slovenia, pp 321–326

  • Collins P, Niqui M, Revol N (2011) A validated real function calculus. Math Comp Sci 5:437–467

    MathSciNet  Article  Google Scholar 

  • Cuninghame-Green RA (1979) Minimax algebra lecture notes in economics and mathematical systems, vol 166. Springer, Berlin

    Google Scholar 

  • Fiedler M, Nedoma J, Ramík J, Rohn J, Zimmermann K (2006) Linear optimization problems with inexact data. Springer, Berlin

    MATH  Google Scholar 

  • Gavalec M, Plavka J, Ponce D (2016) Tolerance types of interval eigenvectors in max-plus algebra. Inform Sci 367-368:14–27

    Article  Google Scholar 

  • Gavalec M, Plavka J, Ponce D (2020) EA/AE-eigenvectors of interval max-min matrices. Mathematics. 8(6)

  • Gavalec M, Plavka J, Tomášková H (2014) Interval eigenproblem in max–min algebra. Linear Algebra Appl 440:24–33

    MathSciNet  Article  Google Scholar 

  • Gavalec M, Plavka J (2003) Strong regularity of matrices in general max–min algebra. Linear Algebra Appl 371:241–254

    MathSciNet  Article  Google Scholar 

  • Litvinov GL, Sobolevskii AN (2001) Idempotent interval ana-lysis and optimization problems. Reliab Comput 7:353–377

    MathSciNet  Article  Google Scholar 

  • Myšková H (2012) On an algorithm for testing t4 solvability of max-plus interval systems. Kybernetika 48(5):924–938

    MathSciNet  MATH  Google Scholar 

  • Myšková H (2012) An iterative algorithm for testing solvability of max-min interval systems. Kybernetika 48(5):879–889

    MathSciNet  MATH  Google Scholar 

  • Myšková H, Plavka J (2013) X-robustness of interval circulant matrices in fuzzy algebra. Linear Algebra Appl 438:2757–2769

    MathSciNet  Article  Google Scholar 

  • Myšková H, Plavka J (2014) The robustness of interval matrices in max-plus algebra. Linear Algebra Appl 445:85–102

    MathSciNet  Article  Google Scholar 

  • Plavka J (2014) The weak robustness of interval matrices in max-plus algebra. Discrete Appl Math 173:92–101

    MathSciNet  Article  Google Scholar 

  • Plavka J (2005) L-parametric Eigenproblem in max-algebra. Discret Appl Math 150:16–28

    MathSciNet  Article  Google Scholar 

  • Plavka J, Sergeev S (2016) Characterizing matrices with X-simple image eigenspace in max-min semiring. Kybernetika 52:497–513

    MathSciNet  MATH  Google Scholar 

  • Plavka J, Sergeev S (2016) X-simple image eigencones of tropical matrices. Linear Algebra and Its Applications 507:169–190

    MathSciNet  Article  Google Scholar 

  • Plavka J, Sergeev S (2018) Reachability of eigenspaces for interval circulant matrices in max-algebra. Linear Algebra and Its Applications 550:59–86

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors are grateful to the referees for their evaluation of the manuscript and for their suggestions and helping comments.

The support of the APVV grant #180373 is gratefully acknowledged.

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Correspondence to Ján Plavka.

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Myšková, H., Plavka, J. Max-plus steady states in discrete event dynamic systems with inexact data. Discrete Event Dyn Syst (2022). https://doi.org/10.1007/s10626-022-00359-3

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  • DOI: https://doi.org/10.1007/s10626-022-00359-3

Keywords

  • Discrete event dynamic systems
  • Max-plus algebra
  • Interval analysis
  • Eigenvector