no. 2
Rayleigh principle for linear hamiltonian systems without controllability
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 501-519.

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.

DOI : 10.1051/cocv/2011104
Classification : 34L05, 34C10, 34L10, 34B09
Mots clés : linear hamiltonian system, Rayleigh principle, self-adjoint eigenvalue problem, proper focal point, conjoined basis, finite eigenvalue, oscillation theorem, controllability, normality, quadratic functional 
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     title = {Rayleigh principle for linear hamiltonian systems without controllability},
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     publisher = {EDP-Sciences},
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Kratz, Werner; Hilscher, Roman Šimon. Rayleigh principle for linear hamiltonian systems without controllability. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 501-519. doi : 10.1051/cocv/2011104. http://archive.numdam.org/articles/10.1051/cocv/2011104/

[1] M. Bohner, O. Došlý and W. Kratz, Sturmian and spectral theory for discrete symplectic systems. Trans. Am. Math. Soc. 361 (2009) 3109-3123. | MR | Zbl

[2] W.A. Coppel, Disconjugacy, Lecture Notes in Mathematics 220. Springer-Verlag, Berlin, Heidelberg (1971). | MR | Zbl

[3] O. Došlý and W. Kratz, Oscillation theorems for symplectic difference systems. J. Difference Equ. Appl. 13 (2007) 585-605. | MR | Zbl

[4] 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. | MR | Zbl

[5] R. Hilscher and V. Zeidan, Riccati equations for abnormal time scale quadratic functionals. J. Differ. Equ. 244 (2008) 1410-1447. | MR

[6] R. Hilscher and V. Zeidan, Nabla time scale symplectic systems. Differ. Equ. Dyn. Syst. 18 (2010) 163-198. | MR | Zbl

[7] W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory. Akademie Verlag, Berlin (1995). | MR | Zbl

[8] 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. | MR | Zbl

[9] W. Kratz, Definiteness of quadratic functionals. Analysis (Munich) 23 (2003) 163-183. | MR | Zbl

[10] 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. | MR | Zbl

[11] W.T. Reid, Ordinary Differential Equations. Wiley, New York (1971). | MR | Zbl

[12] W.T. Reid, Sturmian Theory for Ordinary Differential Equations. Springer-Verlag, New York-Berlin-Heidelberg (1980). | MR | Zbl

[13] R. Šimon Hilscher, and V. Zeidan, Picone type identities and definiteness of quadratic functionals on time scales. Appl. Math. Comput. 215 (2009) 2425-2437. | MR | Zbl

[14] M. Wahrheit, Eigenwertprobleme und Oszillation linearer Hamiltonischer Systeme [Eigenvalue Problems and Oscillation of Linear Hamiltonian Systems]. Ph.D. thesis, University of Ulm, Germany (2006).

[15] M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Difference Equ. 2 (2007) 221-244. | MR

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