Nonlinear Left and Right Eigenvectors for Max-Preserving Maps
Chapter
First Online:
Abstract
It is shown that max-preserving maps (or join-morphisms) on the positive orthant in Euclidean n-space endowed with the component-wise partial order give rise to a semiring. This semiring admits a closure operation for maps that generate stable dynamical systems. For these monotone maps, the closure is used to define suitable notions of left and right eigenvectors that are characterized by inequalities. Some explicit examples are given and applications in the construction of Lyapunov functions are described.
Keywords
Monotone systems Join-morphisms Perron-Frobenius theory Positive eigenvectors Small-gain condition Lyapunov functionsReferences
- 1.Birkoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1973)Google Scholar
- 2.Dirr, G., Ito, H., Rantzer, A., Rüffer, B.S.: Separable Lyapunov functions: constructions and limitations. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2497–2526 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Frobenius, G.: Über Matrizen aus positiven Elementen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, pp. 471–476 (1908)Google Scholar
- 4.Frobenius, G.: Über Matrizen aus positiven Elementen. ii. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, vol. 1909, pp. 514–518 (1909)Google Scholar
- 5.Frobenius, G.: Über Matrizen aus nicht negativen Elementen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, pp. 456–477 (1912)Google Scholar
- 6.Ito, H., Jiang, Z.P., Dashkovskiy, S., Rüffer, B.S.: Robust stability of networks of iISS systems: construction of sum-type Lyapunov functions. IEEE Trans. Autom. Control 58(5), 1192–1207 (2013)MathSciNetCrossRefGoogle Scholar
- 7.Karafyllis, I., Jiang, Z.P.: A vector small-gain theorem for general non-linear control systems. IMA J. Math. Control Inf. 28(3), 309–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Lemmens, B., Nussbaum, R.: Nonlinear Perron-Frobenius Theory. Cambridge Tracts in Mathematics, vol. 189. Cambridge University Press, Cambridge (2012)Google Scholar
- 9.Perron, O.: Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64(1), 1–76 (1907)Google Scholar
- 10.Perron, O.: Zur Theorie der Matrices. Math. Ann. 64(2), 248–263 (1907)Google Scholar
- 11.Rüffer, B.S.: Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean \(n\)-space. Positivity 14(2), 257–283 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Rüffer, B.S., Ito, H.: Sum-separable Lyapunov functions for networks of ISS systems: a gain function approach. In: Proceedings of the 54th IEEE Conference on Decision Control, pp. 1823–1828 (2015)Google Scholar
Copyright information
© Springer International Publishing AG 2017