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Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory

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Abstract

In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Šepitka in Discrete Contin Dyn Syst 39(4):1685–1730, 2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems.

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References

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

    MATH  Google Scholar 

  2. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  3. Bernstein, D.S.: Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  4. Campbell, S. L., Meyer, C.D.: Generalized Inverses of Linear Transformations, Reprint of the 1991 corrected reprint of the 1979 original, Classics in Applied Mathematics, vol. 56. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2009)

  5. Coppel, W.A.: Disconjugacy Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971)

    Google Scholar 

  6. Elyseeva, J.: Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index. J. Math. Anal. Appl. 444(2), 1260–1273 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fabbri, R., Johnson, R., Novo, S., Núñez, C.: Some remarks concerning weakly disconjugate linear Hamiltonian systems. J. Math. Anal. Appl. 380(2), 853–864 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)

    MATH  Google Scholar 

  9. 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 (Balatonfuered, Hungary, 2013), Springer Proceedings in Mathematics & Statistics, vol. 94, pp. 131–159. Springer, Berlin (2014)

  10. Johnson, R., Obaya, R., Novo, S., Núñez, C., Fabbri, R.: Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control Developments in Mathematics, vol. 36. Springer, Cham (2016)

    Book  MATH  Google Scholar 

  11. Kratz, W.: Quadratic Functionals in Variational Analysis and Control Theory. Akademie Verlag, Berlin (1995)

    MATH  Google Scholar 

  12. Kratz, W.: Definiteness of quadratic functionals. Analysis 23(2), 163–183 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Reid, W.T.: Ordinary Differential Equations. Wiley, New York (1971)

    MATH  Google Scholar 

  15. Reid, W.T.: Riccati Differential Equations. Academic Press, New York (1972)

    MATH  Google Scholar 

  16. Reid, W.T.: Sturmian Theory for Ordinary Differential Equations. Springer, New York (1980)

    Book  MATH  Google Scholar 

  17. Speyer, J.L., Jacobson, D.H.: Primer on Optimal Control Theory Advances in Design and Control, vol. 20. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2010)

    Book  MATH  Google Scholar 

  18. Šepitka, P.: Riccati equations for linear Hamiltonian systems without controllability condition. Discrete Contin. Dyn. Syst. 39(4), 1685–1730 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Šepitka, P., Šimon Hilscher, R.: 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 

  21. Šepitka, P., Šimon Hilscher, R.: Principal and antiprincipal solutions at infinity of linear Hamiltonian systems. J. Differ. Equ. 259(9), 4651–4682 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Šepitka, P., Šimon Hilscher, R.: Genera of conjoined bases of linear Hamiltonian systems and limit characterization of principal solutions at infinity. J. Differ. Equ. 260(8), 6581–6603 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Šepitka, P., Šimon Hilscher, R.: Reid’s construction of minimal principal solution at infinity for linear Hamiltonian systems. In: Pinelas, S., Došlá, Z., Došlý, O., Kloeden, P.E. (eds.) Differential and Difference Equations with Applications, Proceedings of the International Conference on Differential & Difference Equations and Applications (Amadora, 2015), Springer Proceedings in Mathematics & Statistics, vol. 164, pp. 359–369, Springer, Berlin (2016)

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Šimon Hilscher, R.: Sturmian theory for linear Hamiltonian systems without controllability. Math. Nachr. 284(7), 831–843 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Šimon Hilscher, R.: On general Sturmian theory for abnormal linear Hamiltonian systems. In: Feng, W., Feng, Z., Grasselli, M., Ibragimov, A., Lu, X., Siegmund, S., Voigt, J. (eds.) Dynamical Systems, Differential Equations and Applications, Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Dresden, 2010), Discrete and Continuous Dynamical Systems, Supplement vol. 2011, pp. 684–691. American Institute of Mathematical Sciences (AIMS), Springfield (2011)

  27. Wahrheit, M.: Eigenvalue problems and oscillation of linear Hamiltonian systems. Int. J. Differ. Equ. 2(2), 221–244 (2007)

    MathSciNet  Google Scholar 

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Acknowledgements

The author is grateful to Professor Roman Šimon Hilscher for consultations regarding the subject of this paper. The author wish to thank an anonymous referee for detailed reading of the paper and valuable suggestions which improved the presentation of the results.

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Correspondence to Peter Šepitka.

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This research was supported by the Czech Science Foundation under Grant GA16-00611S. The revised version of this paper was prepared under the support of the Grant GA19-01246S.

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Šepitka, P. Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory. J Dyn Diff Equat 32, 1139–1155 (2020). https://doi.org/10.1007/s10884-019-09810-w

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