*(English)*Zbl 0855.39018

The author discusses linear Hamiltonian difference systems

where ${x}_{k}$, ${u}_{k}\in {\mathbb{R}}^{n}$, $k\in \overline{J}:=J\cup \{N+1\}$, ${A}_{k}$, ${B}_{k}$, ${C}_{k}$ are $n\times n$-matrices, ${B}_{k}$, ${C}_{k}$ symmetric, ${A}_{k}$ such that ${\tilde{A}}_{k}={(I-{A}_{k})}^{-1}$ 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.

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. *C. D. Ahlbrandt* [J. Math. Anal. Appl. 180, No. 2, 498-517 (1993; Zbl 0802.39005)], *L. H. Erbe* and *P. Yan* [ibid. 167, No. 2, 355-367 (1992; Zbl 0762.39003)].

##### MSC:

39A12 | Discrete version of topics in analysis |

39A10 | Additive difference equations |

93B05 | Controllability |

93C55 | Discrete-time control systems |