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Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions. (English) Zbl 0855.39018
The author discusses linear Hamiltonian difference systems $$\Delta x_k = A_k x_{k + 1} + B_k u_k,\quad \Delta u_k = C_k x_{k + 1} - A^T_k u_k, \quad k \in J : = \{0, 1, \dots, N\}, \tag H$$ 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. 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)].

39A12Discrete version of topics in analysis
39A10Additive difference equations
93C55Discrete-time control systems
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