Skip to main content
SpringerLink
Log in

Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions \(M^\pm \) for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of \(M^+\), then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of \(M^+\) are fundamental to guarantee the solvability of classical minimization problems given by linear–quadratic control processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic, New York (1964)

    MATH  Google Scholar 

  2. Chicone, C., Latuskin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. American Mathematical Society, Providence (1999)

    Book  Google Scholar 

  3. Chow, S.N., Hale, J.: Methods of Bifurcation Theory. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  4. Chow, S.N., Leiva, H.: Exisntence and roughness of the exponential dichotomy for skew-products semiflow in Banach spaces. J. Differ. Equ. 120(2), 429–477 (1995)

    Article  MATH  Google Scholar 

  5. Chow, S.N., Leiva, H.: Unbounded perturbation of the exponential dichotomy for evolution equations. J. Differ. Equ. 129(2), 509–531 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coppel, W.A.: Dichotomies in stability theory. In: Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)

  7. Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. Heath and Co., Boston (1965)

    MATH  Google Scholar 

  8. Ellis, R.: Lectures on Topological Dynamics. Benjamin, New York (1969)

    MATH  Google Scholar 

  9. Fabbri, R., Johnson, R., Núñez, C.: Rotation number for non-autonomous linear Hamiltonian systems I: basic properties. Z. Angew. Math. Phys. 54, 484–502 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fabbri, R., Johnson, R., Núñez, C.: Rotation number for non-autonomous linear Hamiltonian systems II: the Floquet coefficient. Z. Angew. Math. Phys. 54, 652–676 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fabbri, R., Johnson, R., Núñez, C.: On the Yakubovich frequency theorem for linear non-autonomous control processes. Discrete Continuous Dyn. Syst. Ser. A 9(3), 677–704 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fabbri, R., Johnson, R., Novo, S., Núñez, C.: Some remarks concerning weakly disconjugate linear Hamiltonian systems. J. Math. Anal. Appl. 380, 853–864 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fabbri, R., Johnson, R., Novo, S., Núñez, C.: On linear-quadratic dissipative control processes with time-varying coefficients. Discrete Continuous Dyn. Syst. Ser. A 33(1), 193–210 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fabbri, R., Novo, S., Núñez, C., Obaya, R.: Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete Continuous Dyn. Syst. Ser. S 9(4), 1069–1094 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gel’fand, I.M., Lidskiĭ, V.B.: On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients. Am. Math. Soc. Transl. 2(8), 143–181 (1958)

    MathSciNet  Google Scholar 

  17. Gesztesy, F., Tsekanovskii, E.: On matrix-valued Herglotz functions. Math. Nachr. 218, 61–138 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hale, J.K.: Ordinary Differential Equations. Wiley-Interscience, New York (1969)

    MATH  Google Scholar 

  19. Henry, D.: Geometric theory of semilinear parabolic equations. In: Lecture Notes in Mathematics 840. Springer, Berlin (1993)

  20. Hilscher, R.Š., Zemánek, P.: On square integrable solutions and principal and antiprincipal solutions for linear Hamiltonian systems. Ann. Mat. Pura App. https://doi.org/10.1007/s10231-017-0679-7 (2017)

  21. Johnson, R., Novo, S., Núñez, C., Obaya, R.: Uniform weak disconjugacy and principal solutions for linear Hamiltonian systems. In: Recent Advances in Delay Differential and Difference Equations, vol. 94, pp. 131–159. Springer Proceedings in Mathematics & Statistics (2014)

  22. Johnson, R., Obaya, R., Novo, S., Núñez, C., Fabbri, R.: Nonautonomous linear Hamiltonian systems: oscillation, spectral theory and control. In: Developments in Mathematics, vol. 36. Springer (2016)

  23. Johnson, R.: Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients. J. Differ. Equ. 61, 54–78 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Johnson, R.: \(m\)-functions and Floquet exponents for linear differential systems. Ann. Mat. Pura Appl. 147, 211–248 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Johnson, R., Nerurkar, M.: Exponential dichotomy and rotation number for linear Hamiltonian systems. J. Differ. Equ. 108, 201–216 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Johnson, R., Yi, Y.: Hopf bifurcation from non-periodic solutions of differential equations II. J. Differ. Equ. 107, 310–340 (1994)

    Article  MATH  Google Scholar 

  27. Johnson, R., Novo, S., Obaya, R.: An ergodic and topological approach to disconjugate linear Hamiltonian systems. Ill. J. Math. 45, 1045–1079 (2001)

    MathSciNet  MATH  Google Scholar 

  28. Johnson, R., Núñez, C., Obaya, R.: Dynamical methods for linear Hamiltonian systems with applications to control processes. J. Dyn. Differ. Equ. 25(3), 679–713 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Johnson, R., Novo, S., Núñez, C., Obaya, R.: Nonautonomous linear-quadratic dissipative control processes without uniform null controllability. J. Dyn. Differ. Equ. 29, 355–383 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Koosis, P.: Introduction to \(H_p\) spaces. In: London Mathematical Society Lecture Note Series. Cambridge University Press (1980)

  31. Kratz, W.: Quadratic functionals in variational analysis and control theory. Mathematical Topics, vol. 6. Akademie, Berlin (1995)

  32. Massera, J., Schaeffer, J.: Linear Differential Equations and Function Spaces. Academic, New York (1966)

    Google Scholar 

  33. Novo, S., Núñez, C., Obaya, R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equ. 148, 148–185 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Palmer, K.J.: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ. 55(2), 225–256 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Palmer, K.J.: Exponential dichotomies for almost periodic systems. Proc. Am. Math. Soc. 101(2), 293–298 (1987)

    Article  MATH  Google Scholar 

  36. Pliss, V.A., Sell, G.R.: Robustness of exponential dichotomies in infinite dimensional dynamical systems. J. Dyn. Differ. Equ. 11(3), 471–513 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Reid, W.T.: Sturmian theory for ordinary differential equations. In: Applied Mathematical Sciences, vol. 31. Springer, New York (1980)

  38. Reid, W.T.: Principal solutions of nonoscillatory linear differential systems. J. Math. Anal. Appl. 9, 397–423 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rudin, W.: Real and Complex Analysis. McGraw-Hill, Singapore (1987)

    MATH  Google Scholar 

  40. Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems I. J. Differ. Equ. 15, 429–458 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems II. J. Differ. Equ. 22, 478–496 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems III. J. Differ. Equ. 22, 497–522 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sacker, R.J., Sell, G.R.: The spectrum of an invariant submanifold. J. Differ. Equ. 38(2), 135–160 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sacker, R.J., Sell, G.R.: Dichtomies for linear evolutionary equations in Banach spaces. J. Differ. Equ. 113, 17–67 (1994)

    Article  MATH  Google Scholar 

  46. Sell, G.R.: Bifurcation of higher-dimensional tori. Arch. Ration. Mech. Anal. 69(3), 199–230 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  47. Šepitka, P., Hilscher, R.Š.: Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems. J. Dyn. Differ. Equ. 26(1), 57–91 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Šepitka, P., Hilscher, R.Š.: Principal solutions at infinity of given ranks for nonoscillatory linear Hamiltonian systems. J. Dyn. Differ. Equ. 27(1), 137–175 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Šepitka, P., Hilscher, R.Š.: Comparative index and Sturmian theory for linear Hamiltonian systems. J. Differ. Equ. 262(2), 914–944 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  50. Shen, W., Yi, Y.: Almost automorphic and almost periodic dynamics in skew-product semiflows. In: Memoirs of the American Mathematical Society, vol. 647. American Mathematical Society, Providence (1998)

  51. Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. Dyn. Rep. 2, 89–169 (1989)

    MathSciNet  MATH  Google Scholar 

  52. Vanderbauwhede, A., van Gils, S.A.: Center manifolds and contractions on a scale of Banach spaces. J. Funct. Anal. 72(2), 209–224 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yakubovich, V.A.: Linear-quadratic optimization problem and the frequency theorem for periodic systems I. Siberian Math. J. 27(4), 614–630 (1986)

    Article  Google Scholar 

  54. Yakubovich, V.A.: Linear-quadratic optimization problem and the frequency theorem for periodic systems II. Siberian Math. J. 31(6), 1027–1039 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, EII, Universidad de Valladolid, Paseo del Cauce 59, 47011, Valladolid, Spain

    Carmen Núñez & Rafael Obaya

Authors
  1. Carmen Núñez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael Obaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carmen Núñez.

Additional information

Partly supported by MINECO/FEDER (Spain) under project MTM2015-66330-P and by European Commission under project H2020-MSCA-ITN-2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Núñez, C., Obaya, R. Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation. J Dyn Diff Equat 31, 1397–1426 (2019). https://doi.org/10.1007/s10884-017-9637-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-017-9637-8

Keywords

Search

Navigation