Abstract
The analysis of nonautonomous linear Hamiltonian systems with the disconjugacy property is a classical branch of the theory of linear ODEs. One of the most interesting consequences of this property is the existence of principal functions, which constitute an extension of the concept of Weyl functions to many situations where exponential dichotomy is lacking. In this chapter, the disconjugacy property is relaxed to the so-called weak disconjugacy, which can be characterized in terms of the property of nonoscillation of the system. In what follows, a family of systems is considered. It is proved that the so-called uniform weak disconjugacy of all the systems suffices to ensure the existence of the principal functions, whose main properties are then described. And it is shown that this setting is less restrictive than that of disconjugacy. In the rest of the chapter, relations are established between the characteristics of the principal functions for a given family and: (1) the properties of the Lyapunov exponents, (2) the properties of the rotation number, and (3) the existence of exponential dichotomy. The consequences of the analysis presented here will be fundamental in the developments of the remaining chapters of the book.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
W.A. Coppel, Disconjugacy, Lecture Notes in Mathematics 220, Springer-Verlag, Berlin, Heidelberg, New York, 1971.
I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, Ergodic Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
R. Fabbri, R. Johnson, S. Novo, C. Núñez, Some remarks concerning weakly disconjugate linear Hamiltonian systems, J. Math. Anal. Appl. 380 (2011), 853–864.
R. Fabbri, R. Johnson, C. Núñez, On the Yakubovich Frequency Theorem for linear non-autonomous control processes, Discrete Contin. Dynam. Systems, Ser. A 9 (3) (2003), 677–704.
R. Fabbri, R. Johnson, C. Núñez, Disconjugacy and the rotation number for linear, non-autonomous Hamiltonian systems, Ann. Mat. Pura App. 185 (2006), S3–S21.
I.M. Gel’fand, V.B. Lidskiĭ, On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Amer. Mat. Soc. Transl. (2) 8 (1958), 143–181.
P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 1975.
R. Johnson, M. Nerurkar, On null controllability of linear systems with recurrent coefficients and constrained controls, J. Dynam. Differential Equations 4 (2) (1992), 259–273.
R. Johnson, M. Nerurkar, Stabilization and random linear regulator problem for random linear control processes, J. Math. Anal. Appl. 197 (1996), 608–629.
R. Johnson, M. Nerurkar, Controllability, stabilization, and the regulator problem for random differential systems, Mem. Amer. Math. Soc. 646, Amer. Math. Soc., Providence, 1998.
R. Johnson, S. Novo, C. Núñez, R. Obaya, Uniform weak disconjugacy and principal solutions for linear Hamiltonian systems, Recent Advances in Delay Differential and Difference Equations, Springer Proceedings in Mathematics & Statistics 94 (2014), 131–159.
R. Johnson, S. Novo, R. Obaya, Ergodic properties and Weyl M-functions for linear Hamiltonian systems, Proc. Roy. Soc. Edinburgh 130A (2000), 803–822.
R. Johnson, S. Novo, R. Obaya, An ergodic and topological approach to disconjugate linear Hamiltonian systems, Illinois J. Math. 45 (2001), 1045–1079.
R. Johnson, C. Núñez, Remarks on linear-quadratic dissipative control systems Discr. Cont. Dyn. Sys. B, 20 (3) (2015), 889–914.
R. Johnson, C. Núñez, R. Obaya, Dynamical methods for linear Hamiltonian systems with applications to control processes, J. Dynam. Differential Equations 25 (3) (2013), 679–713.
T. Kato, Perturbation Theory for Linear Operators, Corrected Printing of the Second Edition, Springer-Verlag, Berlin, Heidelberg 1995.
W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Mathematical Topics 6, Akademie Verlag, Berlin, 1995.
V.B. Lidskiĭ, Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nauk. SSSR 102 (1955), 877–880. (English translation in: NASA Technical Translation, P-14, 696.)
V.M. Millionščikov, Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients, Diff. Urav. 4 No. 3 (1968), 391–396.
J. Moser, An example of a Schrödinger equation with almost-periodic potential and no-where dense spectrum, Comment. Math. Helv. 56 (1981), 198–224.
W.T. Reid, Principal solutions of nonoscillatory linear differential systems, J. Math. Anal. Appl. 9 (1964), 397–423.
W.T. Reid, A continuity property of principal solutions of linear hamiltonian differential systems, Scripta Math. 29 (1973), 337–350.
W.T. Reid, Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences 31, Springer-Verlag, New York, 1980.
H.H. Schaefer, Topological Vector Spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1970.
I. Schneiberg, Zeros of integrals along trajectories of ergodic systems, Funk. Anal. Appl., 19 (1985), 486–490.
P. Šepitka, R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 26 (1) (2014), 57–91.
P. Šepitka, R. Šimon Hilscher, Principal Solutions at Infinity of Given Ranks for Nonoscillatory Linear Hamiltonian Systems, J. Dynam. Differential Equations 27 (1) (2015), 137–175.
R.E. Vinograd, A problem suggested by N. P. Erugin, Diff. Urav. 11 No. 4 (1975), 632–638.
V.A. Yakubovich, Arguments on the group of symplectic matrices, Mat. Sb. 55 (97) (1961), 255–280 (Russian).
V.A. Yakubovich, Oscillatory properties of the solutions of canonical equations, Amer. Math. Soc. Transl. Ser. 2 42 (1964), 247–288.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Johnson, R., Obaya, R., Novo, S., Núñez, C., Fabbri, R. (2016). Weak Disconjugacy for Linear Hamiltonian Systems. In: Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control. Developments in Mathematics, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-319-29025-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-29025-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29023-2
Online ISBN: 978-3-319-29025-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)