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

The author discusses linear Hamiltonian difference systems

Δx k =A k x k+1 +B k u k ,Δu k =C k x k+1 -A k T u k ,kJ:={0,1,,N},(H)

where x k , u k n , kJ ¯:=J{N+1}, A k , B k , C k are n×n-matrices, B k , C k symmetric, A k such that 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:
39A12Discrete version of topics in analysis
39A10Additive difference equations
93B05Controllability
93C55Discrete-time control systems