Global stabilization in nonlinear discrete systems with time-delay

1. Braverman, E., Liz, E.: Global stabilization of periodic orbits using a proportional feedback control with pulses. Nonlinear Dyn. (in press). Online first at: http://dx.doi.org/10.1007/s11071-011-0160-x

2. Chinchuluun A., Pardalos P.M., Enkhbat R., Tseveendorj I.: Optimization and Optimal Control, pp. 510. Springer, New York, NY (2010)

   Book  Google Scholar 

3. Collet P., Eckmann J.-P.: Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Boston, MA (1980)

   Google Scholar 

4. Cooke K.L., Ivanov A.F.: On the discretization of a delay differential equation. J. Differ. Equ. Appl. 6, 105–119 (2000)

   Article  Google Scholar 

5. Creedy, J., Martin, V.L. (eds.) Chaos and Non-linear Models in Economics. Theory and Applications. p. 228. Aldershot, E. Elgar Pub. (1994)

6. de Melo W., van Strien S.: One-dimensional dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete 3 [Results in Mathematics and Related Areas 3] 25. Springer, Berlin, 1993, 605 pp

7. El-Morshedy H., Liz E.: Globally attracting fixed points in higher order discrete population models. J. Math. Biol. 53, 365–384 (2006)

   Article  Google Scholar 

8. Gandolfo G.: Economic Dynamics, pp. 610. Springer, New York, NY (1996)

   Google Scholar 

9. Goodwin R.M.: Chaotic Economic Dynamics, pp. 137. Oxford University Press, New York, NY (1990)

   Book  Google Scholar 

10. Hirsch M.J., Commander C., Pardalos P.M., Murphey R.: Optimization and Cooperative Control Strategies. Lecture Notes in Control and Information Sciences, vol. 381, pp. 462. Springer, Berlin (2009)

    Book  Google Scholar 

11. Ivanov, A.F., Sharkovsky, A.N.: Oscillations in singularly perturbed delay equations. In: Jones, C.K.R.T., Kirchghaber, U., Walther H.-O. (eds). Dynamics Reported, New Series, vol. 1, pp. 164–224. (1991)

12. Ivanov, A.F., Swishchuk, A.V.: Optimal control of stochastic differential delay equations with application in economics. In: International Journal of Qualitative Theory of Differential Equations and Applications, vol. 2, pp. 201–213. (2008)

13. Khan M.A., Piazza A.: An overview of turnpike theorey: towards the discounted deterministic case. Adv. Math. Econ. 14, 39–67 (2011)

    Article  Google Scholar 

14. Kuang Y.: Delay Differential Equations with Applications in Population Dynamics. Series: Mathematics in Science and Engineering, vol. 191, pp. 398. Academic Press, Boston, MA (2003)

    Google Scholar 

15. Mamedov M.A., Pehlivan S.: Statistical cluster points and turnpike theorem in nonconvex problems. J. Math. Anal. Appl. 256, 686–693 (2001)

    Article  Google Scholar 

16. Mamedov, M.A.: Asymptotical stability of optimal paths in nonconvex problems. In: Pearce C., Hunt, E. (eds.) Optimization: Structure and Applications, Springer, Series Optimization and Its Applications, vol. 32, pp. 95–134 (2009)

17. Mamedov M.A.: Turnpike theorem for continuous-time control systems when optimal stationary point is not unique. Abstr. Appl. Anal. 11, 631–650 (2003)

    Article  Google Scholar 

18. Mammadov M.A., Ivanov A.F.: Asymptotical stability of trajectories in optimal control problems with time delay. In: Barsoum, N., Vasant, P., Habash, R. (‘) Proceedings of the Third Global Conference on Power Control and Optimization, pp. 2–4. Gold Coast, Australia (2010)

    Google Scholar 

19. Mammadov, M.A.: Turnpike Theory: stability of optimal trajectories, In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, vol. XXXIV, p. 4626. (2009) ISBN: 978-0-387-74758-3

20. Pardalos P.M., Yatsenko V.: Optimization and Control of Bilinear Systems. Springer, Series Optimization and Its Applications, vol. 11, pp. 374 (2008)

    Book  Google Scholar 

21. Pehlivan S., Mamedov M.A.: Statistical cluster points and turnpike. Optimization 48, 93–106 (2000)

    Article  Google Scholar 

22. Tkachenko V., Trofimchuk S.: Global stability in difference equations satisfying the generalized Yorke condition. J. Math. Anal. Appl. 303, 173–187 (2005)

    Article  Google Scholar 

23. Sharkovsky, A.N., Kolyada, S.F., Sivak, A.G., Fedorenko, V.V.: Dynamics of One-Dimensional Maps. Ser.: Mathematics and Its Application, vol. 407, p. 261 (1997)

24. Zaslavski A.J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Series: Nonconvex Optimization and Its Applications, vol. 80(XXII), p. 396. (2005)

Download references