Skip to main content
Log in

Semigroups of max-plus linear operators

Semigroup Forum Aims and scope Submit manuscript

An Erratum to this article was published on 04 April 2017


We define strongly continuous max-additive and max-plus linear operator semigroups and study their main properties. We present some important examples of such semigroups coming from non-linear evolution equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions


  1. Akian, M., Gaubert, S., Lakhoua, A.: The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis. SIAM J. Control Optim. 47, 817–848 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akian, M., Gaubert, S., Nussbaum, R.: Uniqueness of the fixed point of nonexpansive semidifferentiable maps. Trans. Amer. Math. Soc (2015). doi:10.1090/S0002-9947-2015-06413-7

  3. Baccelli, F.L., Cohen, G., Olsder, G.-J., Quadrat, J.-P.: Synchronization and Linearity. Wiley, Chichester (1992)

    MATH  Google Scholar 

  4. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff Intern., Publ., Leyden (1976)

    Book  MATH  Google Scholar 

  5. Belleni-Morante, A.: Applied Nonlinear Semigroups, An Introduction. Wiley, Chichester (1998)

    MATH  Google Scholar 

  6. Bressan, A.: Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem. Oxford University Press, New York (2005)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  8. Crandall, M.G.: The semigroup approach to first order quasilinear equations in several space variables. Isr. J. Math. 12, 108–132 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crandall, M.G., Evans, L.C., Lions, P.L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282, 487–502 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Crandall, M.G., Liggett, T.M.: Generation of semi-groups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crandall, M.G., Tartar, L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78, 385–390 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 197. Springer, New York (2000)

    MATH  Google Scholar 

  13. Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes. American Mathematical Society, Providence (2006)

    Book  MATH  Google Scholar 

  14. Gaubert, S.: Théorie des systemes linéaires dans les dioïdes, These, Ecole des Mines de Paris (1992)

  15. Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs. Oxford University Press, New York (1985)

    Google Scholar 

  16. Goldstein, J.A., Nagel, R.: The Evolution of Operator Semigroups, Semigroups of Operators—Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol. 113, pp. 3–25. Springer International Publishing, Cham (2015)

  17. Heidergott, B., Olsder, G.J., van der Woude, J.: Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications. Princeton University Press, Princeton (2005)

    Google Scholar 

  18. Fleming, W.H., Sonner, H.M.: Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics (New York), vol. 25. Springer, New York (1993)

    Google Scholar 

  19. Kandić, M., Peperko, A.: Quotients for idempotent semimodules, Preprint

  20. Kolokoltsov, V.N., Maslov, V.P.: Idempotent Analysis and Applications. Kluwer Academic Publishers, Dordrecht (1997)

    Book  MATH  Google Scholar 

  21. Kraaij, R.: Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces. arXiv:1401.2802v2

  22. Lions, P.-L., Nisio, M.: A uniqueness result for the semigroup associated with the Hamilton-Jacobi-Bellman operator. Proc. Jpn. Acad. Ser. A Math. Sci. 58, 273–276 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Litvinov, G.L., Maslov, V.P. (eds.): Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics, vol. 377. American Mathematical Society, Providence (2005)

    Google Scholar 

  24. Litvinov, G.L., Maslov, V.P., Shpiz, G.B.: Idempotent functional analysis. An algebraic approach. Math. Notes 69(5), 696–729 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. McEneaney, W.M.: Max-Plus Methods for Nonlinear Control and Estimation. Birkhauser, Boston (2006)

    MATH  Google Scholar 

  26. McEneaney, W.M.: A new fundamental solution for differential Riccati equations arising in control. Automatica 44, 920–936 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Morosanu, G.: Nonlinear Evolution Equations and Applications. Springer, Dordrecht (1988)

    MATH  Google Scholar 

  28. Maslov, V.P.: Operatornye metody, (Russian) [Operator methods] Izdat. “Nauka”, Moscow (1973)

  29. Pavel, N.H.: Nonlinear Evolution Operators and Semigroups. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  30. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

Download references


The authors thank J.A. Goldstein, R. Nagel and M. Kandić for useful comments. The first and the second author were supported in part by Grant P1-0222 of the Slovenian Research Agency. The third author was supported by the Bolyai Grant of the Hungarian Academy of Sciences.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Marjeta Kramar Fijavž.

Additional information

Communicated by Markus Haase.

An erratum to this article is available at

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fijavž, M.K., Peperko, A. & Sikolya, E. Semigroups of max-plus linear operators. Semigroup Forum 94, 463–476 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: