Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T07:57:59.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Rayleigh principle for linear Hamiltonian systems withoutcontrollability∗

Published online by Cambridge University Press:  22 July 2011

Werner Kratz and
Roman Šimon Hilscher
Werner Kratz
Affiliation:
Department of Applied Analysis, Faculty of Mathematics and Economics, University of Ulm, 89069 Ulm, Germany. werner.kratz@uni-ulm.de
Roman Šimon Hilscher
Affiliation:
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic; hilscher@math.muni.cz

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we consider linear Hamiltonian differential systems without thecontrollability (or normality) assumption. We prove the Rayleigh principle for thesesystems with Dirichlet boundary conditions, which provides a variational characterizationof the finite eigenvalues of the associated self-adjoint eigenvalue problem. This resultgeneralizes the traditional Rayleigh principle to possibly abnormal linear Hamiltoniansystems. The main tools are the extended Picone formula, which is proven here for thisgeneral setting, results on piecewise constant kernels for conjoined bases of theHamiltonian system, and the oscillation theorem relating the number of proper focal pointsof conjoined bases with the number of finite eigenvalues. As applications we obtain theexpansion theorem in the space of admissible functions without controllability and aresult on coercivity of the corresponding quadratic functional.

Keywords

Linear Hamiltonian systemRayleigh principleself-adjoint eigenvalue problemproper focal pointconjoined basisfinite eigenvalueoscillation theoremcontrollabilitynormalityquadratic functional

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012, pp. 501 - 519
Copyright
© EDP Sciences, SMAI, 2011

References

Bohner, M., Došlý, O. and Kratz, W., Sturmian and spectral theory for discrete symplectic systems. Trans. Am. Math. Soc. 361 (2009) 3109–3123. Google Scholar
W.A. Coppel, Disconjugacy, Lecture Notes in Mathematics 220. Springer-Verlag, Berlin, Heidelberg (1971).
Došlý, O. and Kratz, W., Oscillation theorems for symplectic difference systems. J. Difference Equ. Appl. 13 (2007) 585–605. Google Scholar
J.V. Elyseeva, The comparative index and the number of focal points for conjoined bases of symplectic difference systems in Discrete Dynamics and Difference Equations, in Proceedings of the Twelfth International Conference on Difference Equations and Applications, Lisbon, 2007, edited by S. Elaydi, H. Oliveira, J.M. Ferreira and J.F. Alves. World Scientific Publishing Co., London (2010) 231–238.
Hilscher, R. and Zeidan, V., Riccati equations for abnormal time scale quadratic functionals. J. Differ. Equ. 244 (2008) 1410–1447. Google Scholar
Hilscher, R. and Zeidan, V., Nabla time scale symplectic systems. Differ. Equ. Dyn. Syst. 18 (2010) 163–198. Google Scholar
W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory. Akademie Verlag, Berlin (1995).
Kratz, W., An oscillation theorem for self-adjoint differential systems and the Rayleigh principle for quadratic functionals. J. London Math. Soc. 51 (1995) 401–416. Google Scholar
Kratz, W., Definiteness of quadratic functionals. Analysis (Munich) 23 (2003) 163–183. Google Scholar
Kratz, W., Šimon Hilscher, R., and Zeidan, V., Eigenvalue and oscillation theorems for time scale symplectic systems. Int. J. Dyn. Syst. Differ. Equ. 3 (2011) 84–131. Google Scholar
W.T. Reid, Ordinary Differential Equations. Wiley, New York (1971).
W.T. Reid, Sturmian Theory for Ordinary Differential Equations. Springer-Verlag, New York-Berlin-Heidelberg (1980).
Hilscher, R. Šimon, and Zeidan, V., Picone type identities and definiteness of quadratic functionals on time scales. Appl. Math. Comput. 215 (2009) 2425–2437. Google Scholar
M. Wahrheit, Eigenwertprobleme und Oszillation linearer Hamiltonischer Systeme [Eigenvalue Problems and Oscillation of Linear Hamiltonian Systems]. Ph.D. thesis, University of Ulm, Germany (2006).
Wahrheit, M., Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Difference Equ. 2 (2007) 221–244. Google Scholar