Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

# Advanced Search

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0855.39018
Bohner, Martin
Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions.
(English)
[J] J. Math. Anal. Appl. 199, No.3, 804-826 (1996). ISSN 0022-247X

The author discusses linear Hamiltonian difference systems $$\Delta x_k = A_k x_{k + 1} + B_k u_k,\quad \Delta u_k = C_k x_{k + 1} - A^T_k u_k, \quad k \in J : = \{0, 1, \dots, N\}, \tag H$$ where $x_k$, $u_k \in \bbfR^n$, $k \in \overline J : = J \cup \{N + 1\}$, $A_k$, $B_k$, $C_k$ are $n \times n$-matrices, $B_k$, $C_k$ symmetric, $A_k$ such that $\widetilde A_k = (I - A_k)^{-1}$ exist. For the controllable system (H) the extended Reid Roundabout Theorem is proved; that is equivalence of: a) positivity of some quadratic functional, b) disconjugacy, c) absence of focal points in the principal solution, d) Riccati condition. A particular case of this result, with boundary conditions $x_0 = x_N = 0$, is considered separately. \par To get the main result, a discrete version of Picone's identity is proved, also several definitions like focal points or generalized zeros of vector-valued functions are introduced. Without assumption of nonsingularity of the matrix $B_k$ the presented theory includes discrete Sturm-Liouville equations of higher order. Various interconnections inside the theory and relations with earlier results are widely discussed. See also e.g. {\it C. D. Ahlbrandt} [J. Math. Anal. Appl. 180, No. 2, 498-517 (1993; Zbl 0802.39005)], {\it L. H. Erbe} and {\it P. Yan} [ibid. 167, No. 2, 355-367 (1992; Zbl 0762.39003)].
[J.Popenda (Poznań)]
MSC 2000:
*39A12 Discrete version of topics in analysis
39A10 Difference equations
93B05 Controllability
93C55 Discrete-time control systems

Keywords: Reid roundabout theorem; linear Hamiltonian difference systems; controllable system; disconjugacy; Riccati condition; discrete Sturm-Liouville equations

Citations: Zbl 0802.39005; Zbl 0762.39003

Login Username: Password:

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences
Published by Springer-Verlag | Webmaster