On the Yakubovich frequency theorem for linear non-autonomous control processes

Discrete and Continuous Dynamical Systems

Volumn 9, Issue 3, 2003, Pages 677-704

  Fabbri, Roberta ^a   Johnson, Russell ^a   Núñez, Carmen ^b  

^a UNIVERSITY OF FLORENCE   (Italy)

^b UNIVERSITY OF VALLADOLID   (Spain)

Author keywords

Exponential dichotomy; Frequency Theorem; Rotation number

Indexed keywords

EID: 0037512393     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2003.9.677     Document Type: Article

References (35)

1. Arnold, V. (1969): On a characteristic class entering in a quantum condition, Funk. Anal. Appl., 1, 1-14.
2. Arnold, V. (1985): The Sturm theorems and symplectic geometry, Func. Anal. Appl., 19, 251-259.
3. Anosov, D. (1974): Existence of smooth ergodic flows on smooth manifolds, Math. USSR Izvestia, 8, 525-552.
4. Atkinson, F. (1964): Discrete and Continuous Boundary Value Problems, Academic Press, New York.
5. Basin, M., Pinsky, M. (2002): Control of Kalman-like filters using impulse and continuous feedback design, Discr. Cont. Dyn. Sys. Series B, 2, 169-184.
6. Bebutov, M. (1941): On dynamical systems in the space of continuous functions, Boll. Moskov. Univ. Matematica, 1-52.
7. Choquet, G. (1969): Lectures on Analysis, vol.I, II, III. Benjamin N.Y.
8. Coppel, A. (1978): Dichotomies in Stability Theory, Lecture Notes in Mathematics, 69, Springer-Verlag, Berlin.
9. Ellis, R. (1967): Lectures on Topological Dynamics, Benjamin, New York.
10. Fabbri, R., Johnson, R., Núñez, C.: Rotation number for non-autonomous linear Hamiltonian systems I: basic properties, to appear in Zeit. angew. Math. Phys..
11. Fabbri, R., Johnson, R., Núñez, C.: Rotation number for non-autonomous linear Hamiltonian systems II: the Floquet coefficient, to appear in Zeit. angew. Math. Phys..
12. Gel'fand, I., Lidskii, V. (1955): On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Russian Math. Surveys, 10, 3-40.
13. S. Helgason, S. (1978): Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York.
14. Johnson, R. (1987): m-functions and Floquet exponents for linear differential systems, Ann. Mat. Pura Appl., 147, 211-248.
15. Johnson, R., Nerurkar, M. (1994): Exponential dichotomy and rotation number for linear Hamiltonian systems, Jour. Diff. Eqns., 108, 201-216.
16. Johnson, R., Nerurkar, M. (1996): Stabilization and random linear regulator problem for random linear control processes, Jour. Math. Anal. Appl., 197, 608-629.
17. Johnson, R., Palmer, K. Sell, G. (1987): Ergodic properties of linear dynamical systems, SIAM Jour. Math. Anal., 18, 1-33.
18. Kato, T. (1966): Perturbation Theory for Linear Operators, 1st ed., Springer-Verlag, Berlin.
19. Lidskii, V. (1955): Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nauk. SSSR, 102, 877-880.
20. Nemytskii, V. Stepanov, V. (1960): Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton.
21. Novo, S., Núñez, C., Obaya, R. (1998): Ergodic properties and rotation number for linear Hamiltonian systems, Jour. Diff. Eqns, 148, 148-185.
22. Oseledets, V. (1968): A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19, 197-231.
23. Phelps, R. (1966): Lectures on Choquet Theory, American Book Co., New York.
24. Sacker, R.J., Sell, G.R. (1974): Dichotomies and invariant splittings for linear differential equations I, Jour. Diff. Eqns., 15, 429-458.
25. Sacker, R.J., Sell, G.R. (1978): A spectral theory for linear differential systems, Jour. Diff. Eqns., 27, 320-358.
26. Schneiberg, I. (1985): Zeros of integrals along trajectories of ergodic systems, Funk. Anal. Appl., 19, 486-490.
27. Schwarzmann, S. (1957): Asymptotic cycles, Ann. of Math., 66, 270-284.
28. Seidmann, T., Li, H. (2001): A note on stabilization with saturating feedback, Discr. Cont. Dyn. Sys., 7, 319-328.
29. Selgrade, J. (1975): Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., 203, 359-390.
30. Yakubovich, V. (1961): Arguments on the group of symplectic matrices, Math. Sb., 55, 255-280 (Russian).
31. Yakubovich, V. (1964): Oscillatory properties of the solutions of canonical equations, Amer. Math. Soc. Transl. Ser. 2, 42, 247-288.
32. Yakubovich, V. (1977) Contribution to the abstract theory of optimal control I, Sib. Math. Zhur., 18, 685-707.
33. Yakubovich, V. (1986): Linear-quadratic optimization problem and the frequency theorem for periodic systems I, Siber. Math. Jour., 27, 614-630.
34. Yakubovich, V. (1988): Dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part, Sys. Contr. Lett., 11, 221-228.
35. Yakubovich, V. (1990): Linear-quadratic optimization problem and the frequency theorem for periodic systems II, Siber. Math. Jour., 31, 1027-1039.