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A transformation for symplectic systems and the definition of a focal point. (English) Zbl 1049.93056

Linear time-varying discrete-time systems with system matrices having the symplectic property are considered. State variables are collected into a \(2n\times n\) block matrix instead of the usual \(n\)-dimensional vector. Although no mention of potential applications is given, such system structures are useful in fast calculation of the solutions to Riccati difference equations in the form of so-called “doubling algorithms”. A related Riccati operator is defined for the matrix difference equation and used to state Sturm’s separation theorems.

MSC:

93C55 Discrete-time control/observation systems
39A12 Discrete version of topics in analysis
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
93B40 Computational methods in systems theory (MSC2010)
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[1] Nelson, P.; Ray, A.; Wing, G., On the effectiveness of the inverse Riccati transformation in the matrix case, J. Math. Anal. Appl., 65, 201-210 (1978) · Zbl 0392.65035
[2] Bohner, M.; Došlý, O., Disconjugacy and transformations for symplectic systems, Rocky Mountain J. Math., 27, 3, 707-743 (1997) · Zbl 0894.39005
[3] Bohner, M., Riccati matrix difference equations and linear Hamiltonian difference system, Dynam. Conlin. Discrete Impuls, Systems, 2, 2, 147-159 (1996)
[4] Kratz, W., Discrete oscillation, J. Difference Equations and Appl., 9, 1, 135-147 (2003) · Zbl 1038.39010
[5] Golub, G.; Van Loan, C., Matrix Computations (1996), The Johns Hopkins University Press · Zbl 0865.65009
[6] Zelikin, M., Uniform Spaces and the Riccati Equation in Calculus of Variations (1998), Factorial: Factorial Moscow · Zbl 0951.49001
[7] Eliseeva, Y., An algorithm for solving the matrix difference Riccati equation, Comp. Math. and Math. Phys., 39, 2, 187-194 (1999)
[8] Taufer, J., On factorization method, Aplikace Matematiky, 11, 427-451 (1966) · Zbl 0149.04803
[9] Eliseeva, Y., On an algorithm for solving the symplectic matrix Riccati equation, Moscow University Comp. Math. Cyber., 2, 14-19 (1990) · Zbl 0704.34032
[10] Gantmacher, F., (The Theory of Matrices, Volume 1 (1959), Chelsea Publishing: Chelsea Publishing New York) · Zbl 0085.01001
[11] Bohner, M., Discrete Sturmian theory, Math. Inequal. Appl., 1, 3, 375-383 (1998) · Zbl 0907.39019
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