# Rayleigh principle for linear Hamiltonian systems without controllability∗

Werner Kratz; Roman Šimon Hilscher

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 2, page 501-519
- ISSN: 1292-8119

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topKratz, Werner, and Hilscher, Roman Šimon. "Rayleigh principle for linear Hamiltonian systems without controllability∗." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 501-519. <http://eudml.org/doc/276370>.

@article{Kratz2012,

abstract = {In this paper we consider linear Hamiltonian differential systems without the
controllability (or normality) assumption. We prove the Rayleigh principle for these
systems with Dirichlet boundary conditions, which provides a variational characterization
of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result
generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian
systems. The main tools are the extended Picone formula, which is proven here for this
general setting, results on piecewise constant kernels for conjoined bases of the
Hamiltonian system, and the oscillation theorem relating the number of proper focal points
of conjoined bases with the number of finite eigenvalues. As applications we obtain the
expansion theorem in the space of admissible functions without controllability and a
result on coercivity of the corresponding quadratic functional. },

author = {Kratz, Werner, Hilscher, Roman Šimon},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Linear Hamiltonian system; Rayleigh principle; self-adjoint eigenvalue problem; proper focal point; conjoined basis; finite eigenvalue; oscillation theorem; controllability; normality; quadratic functional; linear Hamiltonian system},

language = {eng},

month = {7},

number = {2},

pages = {501-519},

publisher = {EDP Sciences},

title = {Rayleigh principle for linear Hamiltonian systems without controllability∗},

url = {http://eudml.org/doc/276370},

volume = {18},

year = {2012},

}

TY - JOUR

AU - Kratz, Werner

AU - Hilscher, Roman Šimon

TI - Rayleigh principle for linear Hamiltonian systems without controllability∗

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2012/7//

PB - EDP Sciences

VL - 18

IS - 2

SP - 501

EP - 519

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

LA - eng

KW - Linear Hamiltonian system; Rayleigh principle; self-adjoint eigenvalue problem; proper focal point; conjoined basis; finite eigenvalue; oscillation theorem; controllability; normality; quadratic functional; linear Hamiltonian system

UR - http://eudml.org/doc/276370

ER -

## References

top- M. Bohner, O. Došlý and W. Kratz, Sturmian and spectral theory for discrete symplectic systems. Trans. Am. Math. Soc.361 (2009) 3109–3123. Zbl1167.39009
- W.A. Coppel, Disconjugacy, Lecture Notes in Mathematics220. Springer-Verlag, Berlin, Heidelberg (1971).
- O. Došlý and W. Kratz, Oscillation theorems for symplectic difference systems. J. Difference Equ. Appl.13 (2007) 585–605. Zbl1124.39006
- 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. Zbl1217.39007
- R. Hilscher and V. Zeidan, Riccati equations for abnormal time scale quadratic functionals. J. Differ. Equ.244 (2008) 1410–1447. Zbl05255887
- R. Hilscher and V. Zeidan, Nabla time scale symplectic systems. Differ. Equ. Dyn. Syst.18 (2010) 163–198. Zbl1214.34091
- W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory. Akademie Verlag, Berlin (1995). Zbl0842.49001
- W. Kratz, An oscillation theorem for self-adjoint differential systems and the Rayleigh principle for quadratic functionals. J. London Math. Soc.51 (1995) 401–416. Zbl0823.93025
- W. Kratz, Definiteness of quadratic functionals. Analysis (Munich)23 (2003) 163–183. Zbl1070.49014
- W. Kratz, R. Šimon Hilscher, and V. Zeidan, Eigenvalue and oscillation theorems for time scale symplectic systems. Int. J. Dyn. Syst. Differ. Equ.3 (2011) 84–131. Zbl1218.34039
- W.T. Reid, Ordinary Differential Equations. Wiley, New York (1971). Zbl0212.10901
- W.T. Reid, Sturmian Theory for Ordinary Differential Equations. Springer-Verlag, New York-Berlin-Heidelberg (1980). Zbl0459.34001
- R. Šimon Hilscher, and V. Zeidan, Picone type identities and definiteness of quadratic functionals on time scales. Appl. Math. Comput.215 (2009) 2425–2437. Zbl1178.93094
- M. Wahrheit, Eigenwertprobleme und Oszillation linearer Hamiltonischer Systeme [Eigenvalue Problems and Oscillation of Linear Hamiltonian Systems]. Ph.D. thesis, University of Ulm, Germany (2006).
- M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Difference Equ.2 (2007) 221–244.

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