Exponential stability for discrete time linear equations defined by positive operators

Integral Equations and Operator Theory

Volumn 54, Issue 4, 2006, Pages 465-493

  Dragan, Vasile ^a   Morozan, Toader ^a  

^a INSTITUTE OF MATHEMATICS   (Romania)

Author keywords

Discrete time linear equations; Exponential stability; Positive operators

Indexed keywords

EID: 33645544407     PISSN: 0378620X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00020-005-1371-7     Document Type: Article

References (35)

1. L. Arnold, Stochastic Differential Equations; Theory and Applications; Wiley: New York, 1974.
2. A. Berman, M. Neumann, R.J.Stern, Nonnegative matrices in Dynamic Systems, John Waley and Sons, New York, 1989.
3. ∞ type control for discrete-time stochastic systems, Int. J. Robust Nonlinear Control, 9, (1999), 923-948.
4. O.L.V. Costa, M.D. Fragoso, Necessary and sufficient conditions for mean square stability of discrete-time linear systems subject to Markovian jumps, Proc. 9-th Int. Symp. on Math. Theory of Networks and Systems, pp. 85-86, Kobe, Japan, 1991.
5. O.L.V. Costa, M.D. Fragoso, Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl., 179, 2, (1993), 154-178.
6. R.F. Curtain, Stability of Stochastic Dynamical Systems: Lecture Notes in Mathematics; Springer Verlag, vol. 294, 1972.
7. T. Damm, D. Hinrichsen, Newton's method for a rational matrix equation occuring in stochastic control, Linear Algebra Appl. 332/334, (2001), 81-109.
8. T. Damm, D. Hinrichsen, Newton's method for concave operators with resolvent positive derivatives in ordered Banach spaces, Linear Algebra Appl., 363, (2003), 43-64.
9. G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.
10. J.L.Doob, Stochastic processes, Wiley, New-York, 1967.
11. V. Dragan, G. Freiling, A. Hochhaus, T. Morozan, A class of nonlinear differential equations on the space of symmetric matrices,Electron. J. Diff. Eqns., Vol. 2004(2004), No. 96, 1-48.
12. Y. Fang, K. Loparo, Stochastic stability of jump linear systems, IEEE Trans. Aut. Control, 47, 7, (2002), 1204-1208.
13. X. Feng, K. Loparo, Stability of linear Markovian jump systems, Proc. 29-th IEEE Conf. Decision Control, 1408, Honolulu, Hawaii, 1990.
14. X. Feng, K. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Aut. Control, 37, 1, (1992), 38-53.
15. M.D. Fragoso, O.L.V. Costa, C.E. de Souza, A new approach to linearly perturbed Riccati equations arising in stochastic control, Appl. Math. Optim., 37, (1998), 99-126.
16. G. Freiling, A. Hochhaus, Properties of the solutions of rational matrix difference equations. Advances in difference equations.IV, Comput. Math. Appl.,45, (2003), 1137-1154.
17. A. Halanay, D. Wexler, Qualitative Theory of Systems with Impulses, Romanian Academy Publishing House, 1968, Russian translation MIR Moskow, 1971.
18. Y. Ji, H.J. Chizeck, X. Feng, K. Loparo, Stability and control of discrete-time jump linear systems, Control Theory and Advances Tech., 7, 2, (1991), 247-270.
19. R.Z. Khasminskii, Stochastic Stability of Differential Equations: Sythoff and Noordhoff: Alpen aan den Ryn, 1980.
20. M.A. Krasnosel'skij, J.A. Lifshits, A.V.Sobolev, Positive Linear Systems- The Method of Positive Operators, volume 5 of Sigma Series in Applied Mathematics, Heldermann Verlag, Berlin, 1989.
21. R. Krtolica, U. Ozguner, H. Chan, H. Goktas, J. Winkelman and M. Liubakka, Stability of linear feedback systems with random communication delays, Proc. 1991 ACC, Boston, MA., June 26-28, 1991.
22. M.G. Krein, R. Rutman, Linear operators leaving invariant a cone in a Banach space, American Math. Soc. Translations, Ser.1, 10, (1962),199-325
23. (originally Uspehi Mat. Nauk (N.S.) 3, 3-95 (1948)).
24. M. Mariton, Jump linear systems in Automatic control, Marcel Dekker, New-York, 1990.
25. T. Morozan, Stability and control of some linear discrete-time systems with jump Markov disturbances, Rev. Roum. Math. Rures et Appl., 26, 1, (1981), 101-119.
26. T. Morozan, Optimal stationary control for dynamic systems with Markovian perturbations,Stochastic Anal. and Appl., 1, 3, (1983), 299-325.
27. T. Morozan, Stability and control for linear discrete-time systems with Markov perturbations, Rev. Roum. Math. Pures et Appl., 40, 5-6,(1995),471-494.
28. T. Morozan, Stabilization of some stochastic discrete-time control systems. Stochastic Anal. and Appl., 1, 1, (1983), 89-116.
29. T. Morozan, Stability radii of some discrete-time systems with independent random perturbations, Stochastic Anal. and Appl., 15, 3,(1997), 375-386.
30. T. Morozan, Discrete-time Riccati equations connected with quadratic control for linear systems with independent random pertrubations, Rev. Roum.Math. Pures et Appl., 37, 3, (1992),233-246.
31. A.C.M. Ran, M.C.B. Reurings, The symmetric linear matrix equation, The Electronic Journal of Linear Algebra, 9, (2002), 93-107.
32. H. Schneider, Positive operators and an Inertia Theorem, Numerische Mathematik, 7, (1965), 11-17.
33. W.H. Wonham, Random Differential Equations in Control Theory. In Probabilistic Methods in Applied Mathematics; Barucha-Reid, A.T., Ed.; Academic Press: New York, 2, (1970), 131-212.
34. J. Yong, X.Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
35. J. Zabczyk, Stochastic control of discrete-time systems, Control Theory and Topics in Funct. Analysis, 3, IAEA, Vienna, 1976.