Structure of extremal trajectories of discrete linear systems and the finiteness conjecture

Automation and Remote Control

Volumn 68, Issue 1, 2007, Pages 174-209

  Kozyakin, V S ^a  

^a INSTITUTE FOR INFORMATION TRANSMISSION PROBLEMS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

COMBINATORIAL MATHEMATICS; CONTROL THEORY; SET THEORY;

DYNAMICAL SYSTEMS; FINITENESS CONJECTURES; MATRIX PRODUCTS; SQUARE MATRICES;

LINEAR SYSTEMS;

EID: 33846681095     PISSN: 00051179     EISSN: None     Source Type: Journal    
DOI: 10.1134/S0005117906040171     Document Type: Article

References (21)

1. Barabanov, N.E., Lyapunov Indicator of Discrete Inclusions. I-III, Autom. Remote Control, 1988, vol. 49, no. 2, pp. 152-157; vol. 49, no. 3, pp. 283-287; vol. 49, no. 5, pp. 558-565.
2. Berger, M.A. and Wang, Y., Bounded Semigroups of Matrices, Lin. Algebra Appl., 1992, vol. 166, pp. 21-27.
3. Blondel, V.D., Theys, J., and Vladimirov, A.A., Switched Systems That Are periodically Stable May Be Unstable, in Proc. of the Symposium MTNS, Notre-Dame, USA, 2002.
4. Blondel, V.D., Theys, J., and Vladimirov, A.A., An Elementary Counterexample to the Finiteness Conjecture, SIAM J. Matrix Analysis, 2003, vol. 24, no. 4, pp. 963-970.
5. Bousch, T. and Mairesse, J., Asymptotic Height Optimization for Topical IFS, Tetris Heaps, and the Finiteness Conjecture, J. Am. Math. Soc., 2002, vol. 15, no. 1, pp. 77-111.
6. Brette, R., Rotation Numbers of Discontinuous Orientation-preserving Circle Maps, Set-Valued Analysis, 2003, vol. 11, no. 4, pp. 359-371.
7. Heil, C. and Strang, G., Continuity of the Joint Spectral Radius: Application to Wavelets, in Linear Algebra for Signal Processing, Bojanczyk, A. and Cybenko G., Eds., Minneapolis: IMA, 1992.
8. Kozyakin, V.S., Algebraic Unsolvability of Problem of Absolute Stability of Desynchronized Systems, Autom. Remote Control, 1990, vol. 51, no. 6, pp. 754-759.
9. Kozyakin, V.S., Sturmian Sequences Generated by Order Preserving Circle Maps, Preprint of Boole Centre for Research in Informatics, University College Cork, National University of Ireland, Cork, 2003, no. 11.
10. Kozyakin, V.S., Discontinuous Order Preserving Circle Maps Versus Circle Homeomorphisms, Preprint of Boole Centre for Research in Informatics, University College Cork, National University of Ireland, Cork, 2003, no. 12.
11. Kozyakin, V.S., Extremal Norms, Discontinuous Circle Maps and a Counterexample to the Finiteness Conjecture, Information Processes, 2005, vol. 5, no. 4, pp. 301-335.
12. Kozyakin, V.S., Proof of a Counterexample to the Finiteness Conjecture in the Spirit of the Theory of Dynamical Systems, Preprint of Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, 2005, no. 105.
13. Kozyakin, V.S. and Pokrovskii, A.V., The Role of Controllability-type Properties in the Study of the Stability of Desynchronized Dynamical Systems, Soviet Phys. Dokl., 1992, vol. 37, no. 5, pp. 213-215.
14. Kozyakin, V.S. and Pokrovskii, A.V., Estimates of Amplitudes of Transient Regimes in Quasi-controllable Discrete Systems, CADSEM Report 96-005, Deakin University, Geelong, Australia, 1996.
15. Kozyakin, V.S. and Pokrovskii, A.V., Quasi-controllability and Estimation of the Amplitudes of Transient Regimes in Discrete Systems, Trans. Russ. Acad. Natural Sci., ser. MMMIC, 1997, vol. 1, no. 3, pp. 128-150.
16. Kuratowski, K, Topology, New York: Academic, 1966, vol. 1.
17. Lagarias, J.C. and Wang, Y., The Finiteness Conjecture for the Generalized Spectral Radius of a Set of Matrices, Lin. Algebra Appl., 1995, vol. 214, pp. 17-42.
18. Miloslavljevich, C., General Conditions for the Existence of Quasisliding Mode on the Switching Hyperplane in Discrete Variable Structure Systems, Autom. Remote Control, 1985, vol. 46, no. 3, pp. 307-314.
19. Misawa, E.A., Discrete-time Sliding Mode Control: The Linear Case, ASME J. Dynamical Syst., Measur., Control, 1997, vol. 119, no. 4, pp. 819-821.
20. Wirth, F., The Generalized Spectral Radius and Extremal Norms, Lin. Algebra Appl., 2002, vol. 342, pp. 17-40.
21. Wirth, F., On the Structure of the Set of Extremal Norms of a Linear Inclusion, in Proc. of the 44th IEEE Conf. on Decision and Control, and the Eur. Control Conf., Seville, Spain, 2005, pp. 3019-3024.