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Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions. (English) Zbl 0855.39018

The author discusses linear Hamiltonian difference systems

${\Delta }{x}_{k}={A}_{k}{x}_{k+1}+{B}_{k}{u}_{k},\phantom{\rule{1.em}{0ex}}{\Delta }{u}_{k}={C}_{k}{x}_{k+1}-{A}_{k}^{T}{u}_{k},\phantom{\rule{1.em}{0ex}}k\in J:=\left\{0,1,\cdots ,N\right\},\phantom{\rule{2.em}{0ex}}\left(\mathrm{H}\right)$

where ${x}_{k}$, ${u}_{k}\in {ℝ}^{n}$, $k\in \overline{J}:=J\cup \left\{N+1\right\}$, ${A}_{k}$, ${B}_{k}$, ${C}_{k}$ are $n×n$-matrices, ${B}_{k}$, ${C}_{k}$ symmetric, ${A}_{k}$ such that ${\stackrel{˜}{A}}_{k}={\left(I-{A}_{k}\right)}^{-1}$ 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