Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales
In this paper we study qualitative properties of the socalled symplectic dynamic system (S) z δ A =Stz on an arbitrary time scale T, providing a unified theory for discrete symplectic systems and differential linear Hamiltonian systems . We define dis-conjugacy (no focal points) for conjoined bases...
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| Published in | Journal of difference equations and applications Vol. 7; no. 2; pp. 265 - 295 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Gordon and Breach Science Publishers
01.01.2001
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1023-6198 1563-5120 |
| DOI | 10.1080/10236190108808273 |
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| Abstract | In this paper we study qualitative properties of the socalled symplectic dynamic system (S) z
δ
A =Stz on an arbitrary time scale
T, providing a unified theory for discrete symplectic systems
and differential linear Hamiltonian systems
. We define dis-conjugacy (no focal points) for conjoined bases of (S) and prove, under a certain minimal normality assumption, that disconjugacy of (S) on the interval under consideration is equival ent to the positivity of the associated quadratic functional. Such statement is commonly called Jacobi condition. We discuss also the solvability of the corresponding Riccati matrix equation and transformations. This work may be regarded as a generalization of the results recently obtained by the second author for linear Hamiltonian systems on time scales. |
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| AbstractList | In this paper we study qualitative properties of the socalled symplectic dynamic system (S) z
δ
A =Stz on an arbitrary time scale
T, providing a unified theory for discrete symplectic systems
and differential linear Hamiltonian systems
. We define dis-conjugacy (no focal points) for conjoined bases of (S) and prove, under a certain minimal normality assumption, that disconjugacy of (S) on the interval under consideration is equival ent to the positivity of the associated quadratic functional. Such statement is commonly called Jacobi condition. We discuss also the solvability of the corresponding Riccati matrix equation and transformations. This work may be regarded as a generalization of the results recently obtained by the second author for linear Hamiltonian systems on time scales. |
| Author | Hilscher, Roman Dořlý, Ondřrej |
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| References | Ben-Israel A. (CIT0009) 1974 Bohner M. (CIT0015) Hilscher R. (CIT0028) Anderson D.R. (CIT0007) 1999; 8 CIT0010 Bohner M. (CIT0012) 1997; 12 Bohner M. (CIT0011) 1996; 2 Bohner M. (CIT0013) 1998; 1 Hilger S. (CIT0026) 1990; 18 Anderson D.R. (CIT0006) 1997; 7 Coppel W.A (CIT0018) 1971; 220 CIT0014 Hilscher R. (CIT0031) 1998 CIT0035 Kaymakcalan B. (CIT0033) 1996 CIT0017 Hilscher R. (CIT0027) 1999; 8 Došlý O. (CIT0019) 1990; 50 Erbe L. (CIT0023) 1993; 1 Hilscher R. (CIT0029) Reid W.T. (CIT0036) 1971 CIT0021 Došlý O. (CIT0020) 1995; 31 Ahlbrandt C.D. (CIT0004) 1981; 81 Ahlbrandt C.D. (CIT0005) 1996 Kratz W. (CIT0034) 1995 Došlý O. (CIT0022) 1999; 8 Hilscher R. (CIT0030) Aulbach B. (CIT0008) 1988 CIT0003 CIT0025 Hilscher R. (CIT0032) Agarwal R.P. (CIT0001) 1999; 8 CIT0002 Bohner M. (CIT0016) 1999; 8 CIT0024 |
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| Snippet | In this paper we study qualitative properties of the socalled symplectic dynamic system (S) z
δ
A =Stz on an arbitrary time scale
T, providing a unified theory... |
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| SubjectTerms | 1991 Mathematics Subject Classification: 34C10,39A 10,93C70 Disconjugacy Focal point Jacobi condition Linear Hamiltonian system Principal solution Quadratic functional Riccati equation Symplectic system Time scale |
| Title | Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales |
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