Advertisement

Uniform Weak Disconjugacy and Principal Solutions for Linear Hamiltonian Systems

  • Russell Johnson
  • Sylvia Novo
  • Carmen Núñez
  • Rafael ObayaEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 94)

Abstract

The paper analyzes the property of (uniform) weak disconjugacy for nonautonomous linear Hamiltonian systems, showing that it is a convenient replacement for the more restrictive property of disconjugacy. In particular, its occurrence ensures the existence of principal solutions. The analysis of the properties of these solutions provides ample information about the dynamics induced by the Hamiltonian system on the Lagrange bundle.

Keywords

Nonautonomous linear Hamiltonian systems Uniform weak disconjugacy Principal solutions 

Notes

Acknowledgements

Partly supported by MIUR (Italy), by MEC (Spain) under project MTM2012-30860, and by JCyL (Spain) under project VA118A12-1.

References

  1. 1.
    Coppel, W.A.: Disconjugacy. Lecture Notes in Mathematics 220, Springer, Berlin/Heidelberg, New York (1971)Google Scholar
  2. 2.
    Fabbri, R., Johnson, R., Novo, S., Núñez, C.: Some remarks concerning weakly disconjugate linear Hamiltonian systems. J. Math. Anal. Appl. 380, 853–864 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Fabbri, R., Johnson, R., Núñez, C.: Rotation number for non-autonomous linear Hamiltonian systems I: Basic properties. Z. Angew. Math. Phys. 53, 484–502 (2002)Google Scholar
  4. 4.
    Fabbri, R., Johnson, R., Núñez, C.: On the Yakubovich Frequency Theorem for linear non-autonomous control processes. Discrete Contin. Dynam. Systems, Ser. A 9(3), 677–704 (2003)Google Scholar
  5. 5.
    Fabbri, R., Johnson, R., Núñez, C.: Disconjugacy and the rotation number for linear, non-autonomous Hamiltonian systems. Ann. Mat. Pura App. 185, S3–S21 (2006)CrossRefzbMATHGoogle Scholar
  6. 6.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  7. 7.
    Johnson, R., Nerurkar, M.: Exponential dichotomy and rotation number for linear Hamiltonian systems. J. Differ. Equat. 108, 201–216 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Johnson, R., Nerurkar, M.: Controllability, stabilization, and the regulator problem for random differential systems. Mem. American Mathematical Society, vol. 646, American Mathematical Society, Providence, RI (1998)Google Scholar
  9. 9.
    Johnson, R., Novo, S., Obaya, R.: Ergodic properties and Weyl M-functions for linear Hamiltonian systems. Proc. Roy. Soc. Edinburgh 130A, 803–822 (2000)MathSciNetGoogle Scholar
  10. 10.
    Johnson, R., Novo, S., Obaya, R.: An ergodic and topological approach to disconjugate linear Hamiltonian systems. Illinois J. Math. 45, 1045–1079 (2001)MathSciNetGoogle Scholar
  11. 11.
    Johnson, R., Núñez, C.: Remarks on linear-quadratic dissipative control systems. Discrete Contin. Dynam. Systems, Ser. B, to appear.Google Scholar
  12. 12.
    Johnson, R., Núñez, C., Obaya, R.: Dynamical methods for linear Hamiltonian systems with applications to control processes. J. Dynam. Differ. Equat. 25(3), 679–713 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kato, T.: Perturbation theory for linear operators. Corrected Printing of the Second Edition, Springer, Berlin/Heidelberg (1995)zbMATHGoogle Scholar
  14. 14.
    Lidskiĭ, V.B.: Oscillation theorems for canonical systems of differential equations. Dokl. Akad. Nank. SSSR 102, 877–880 (1955) (English translation in: NASA Technical Translation, P-14, 696)Google Scholar
  15. 15.
    Mañé, R.: Ergodic Theory and Differentiable Dynamics. Springer, Berlin/Heidelberg, New York (1987)CrossRefzbMATHGoogle Scholar
  16. 16.
    Novo, S., Núñez, C., Obaya, R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equat. 148, 148–185 (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Reid, W.T.: Sturmian theory for ordinary differential equations. Applied Mathematical Sciences, vol. 31, Springer, New York (1980)Google Scholar
  18. 18.
    Yakubovich, V.A.: Arguments on the group of symplectic matrices. Mat. Sb. 55 (97), 255–280 (1961) (Russian)MathSciNetGoogle Scholar
  19. 19.
    Yakubovich, V.A.: Oscillatory properties of the solutions of canonical equations. Amer. Math. Soc. Transl. Ser. 2 42, 247–288 (1964)Google Scholar
  20. 20.
    Yakubovich, V.A., Starzhinskii, V.M.: Linear Differential Equations with Periodic Coefficients. John Wiley and Sons, Inc., New York (1975)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Russell Johnson
    • 1
  • Sylvia Novo
    • 2
  • Carmen Núñez
    • 2
  • Rafael Obaya
    • 2
    Email author
  1. 1.Dipartimento di Matematica e InformaticaUniversità di FirenzeFirenzeItaly
  2. 2.Departamento de Matemática AplicadaEscuela de Ingenierías Industriales, Paseo del Cauce 59, Universidad de ValladolidValladolidSpain

Personalised recommendations