Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean n-space

Positivity

Volumn 14, Issue 2, 2010, Pages 257-283

  Rüffer, Björn S ^a  

^a UNIVERSITY OF NEWCASTLE   (Australia)

Author keywords

Comparison functions; Decay sets; Monotone operators and inequalities; Monotone selections; Omega paths; Positive systems

Indexed keywords

EID: 77953476491     PISSN: 13851292     EISSN: 15729281     Source Type: Journal    
DOI: 10.1007/s11117-009-0016-5     Document Type: Article

References (17)

1. Berman A., Plemmons R. J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press (Harcourt Brace Jovanovich Publishers), New York (1979).
2. Dashkovskiy S., Rüffer B. S., Wirth F. R.: An ISS small gain theorem for general networks. Math. Control Signals Syst. 19(2), 93-122 (2007).
3. Dashkovskiy, S., Rüffer, B. S., Wirth, F. R.: Small gain theorems for large scale systems and construction of ISS Lyapunov functions. January 2009 (submitted).
4. Hahn W.: Theorie und Anwendung der direkten Methode von Ljapunov. Ergebnisse der Mathematik und ihrer Grenzgebiete N. F., Heft 22. Springer, Berlin (1959).
5. Hahn, W.: Stability of motion. Translated from the German manuscript by Arne P. Baartz. Die Grundlehren der mathematischen Wissenschaften, Band 138. Springer, New York (1967).
6. Horvath C. D., Lassonde M.: Intersection of sets with n-connected unions. Proc. Am. Math. Soc. 125(4), 1209-1214 (1994).
7. Knaster B., Kuratowski C., Mazurkiewicz S.: Ein Beweis des Fixpunktsatzes für n-dimensionale Simplexe. Fundamenta 14, 132-137 (1929).
8. Krasnosel′skiǐ M. A.: The Operator of Translation along the Trajectories of Differential Equations. American Mathematical Society, Providence (1968).
9. Krasnosel′skiǐ M. A., Zabreǐko P. P.: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984).
10. Kuratowski K.: Introduction to Set Theory and Topology. Pergamon Press, New York (1961).
11. Lassonde M.: Sur le principe KKM. C. R. Acad. Sci. Paris Sér. I Math. 310(7), 573-576 (1990).
12. Lur Y.-Y.: On the asymptotic stability of nonnegative matrices in max algebra. Linear Algebra Appl. 407, 149-161 (2005).
13. Munkres J. R.: Topology: A First Course. Prentice-Hall, Englewood Cliffs (1975).
14. Potrykus, H. G.: An idempotent-Analytic ISS Small Gain Theorem with Applications to Complex Process Models. PhD thesis, The University of Texas at Austin (2002).
15. Potrykus, H. G., Allgöwer, F., Joe Qin, S.: The Character of an Idempotent-analytic Nonlinear Small Gain Theorem. In Positive systems (Rome 2003). Lecture Notes in Control and Information Sciences, vol. 294, pp. 361-368. Springer, Berlin (2003).
16. Rüffer, B. S.: Monotone Systems, Graphs, and Stability of Large-Scale Interconnected Systems. PhD thesis, Fachbereich 3, Mathematik und Informatik, Universität Bremen, Germany, 2007. http://nbn-resolving. de/urn: nbn: de: gbv: 46-diss000109058.
17. Teel, A.: Input-to-state stability and the nonlinear small gain theorem. Chapter 12 in his lecture notes, courtesy due to private communication, June 2005.