Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-20T14:42:56.918Z Has data issue: false hasContentIssue false
  • English
  • Français

An oscillation theorem for self-adjoint differential systems and an index result for corresponding Riccati matrix differential equations

Published online by Cambridge University Press:  24 October 2008

Werner Kratz
Werner Kratz
Affiliation:
Abteilung Mathematik V, Universität Ulm, D-89069 Ulm, Germany e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The main result of this paper is an oscillation theorem on linear self-adjoint differential systems and a corresponding eigenvalue problem. It establishes a formula between the number of focal points of a so-called conjoined basis of the differential system on a given compact interval and the number of eigenvalues which are less than the given eigenvalue parameter. It extends an earlier result of the author and generalizes an oscillation theorem of M. Morse. Among others the proof of the theorem requires a formula on the index of the difference of symmetric solutions of a corresponding Riccati matrix differential equation. This index formula is the other new result presented.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 351 - 361
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baur, G.. An oscillation theorem for the Sturm-Liouville problem with self-adjoint boundary conditions. Math. Nachr. 138 (1988), 189–194.CrossRefGoogle Scholar
[2]Baur, G. and Kratz, W.. A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functions. Rend. Circ. Mat. Palermo (2) 38 (1989), 329–370.CrossRefGoogle Scholar
[3]Birkhoff, G. D.. Existence and oscillation theorem for a certain boundary value problem. Trans. Amer. Math. Soc. 10 (1909), 259–270.CrossRefGoogle Scholar
[4]Gantmacher, F. R.. The theory of matrices, vol. I (Chelsea, 1960).Google Scholar
[5]Klamka, J.. Controllability of dynamical systems (Kluwer Academic Publishers, 1991).Google Scholar
[6]Kratz, W.. The asymptotic behaviour of Riccati matrix differential equations. Asymptotic Anal. 7 (1993), 67–80.CrossRefGoogle Scholar
[7]Kratz, W.. An oscillation theorem for self-adjoint differential systems and the Rayleigh principle for quadratic functionals. J. London Math. Soc. (to appear).Google Scholar
[8]Kratz, W.. An index theorem for monotone matrix-valued functions. SIAM J. Matrix Anal. Appl. 16 (1995) (to appear).CrossRefGoogle Scholar
[9]Kratz, W.. Characterization of strong observability and construction of an observer. Linear Algebra Appl, (to appear).Google Scholar
[10]Kratz, W. and Peyerimhoff, A.. A treatment of Sturni-Liouville eigenvalue problems via Picone's identity. Analysis 5 (1985), 97–152.CrossRefGoogle Scholar
[11]Märker, F. J.. On the asymptotic behaviour of certain Riccati matrix differential equations. Asymptotic Anal. 6 (1993), 295–314.CrossRefGoogle Scholar
[12]Marcus, M. and Minc, H.. A survey of matrix theory and matrix inequalities (Allyn and Bacon, 1964).Google Scholar
[13]Morse, M.. The calculus of variations in the large (AMS Colloquium Publication 18, 1934).CrossRefGoogle Scholar
[14]Morse, M.. Variational Analysis: Critical extremals and Sturmian extensions (Wiley, 1973).Google Scholar
[15]Reid, W. T.. Ordinary differential equations (Wiley, 1971).Google Scholar
[16]Reid, W. T.. Riccati differential equations (Academic Press, 1972).Google Scholar
[17]Wonham, W. M.. Linear multivariable control: a geometric approach, 3rd ed. (Springer, 1985).CrossRefGoogle Scholar