Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2008 (2008), Article ID 295817, 14 pages
http://dx.doi.org/10.1155/2008/295817
Research Article

On the Continuity Properties of the Attainable Sets of Nonlinear Control Systems with Integral Constraint on Controls

Department of Mathematics, Anadolu University, Eskisehir 26470, Turkey

Received 11 June 2007; Revised 30 August 2007; Accepted 6 November 2007

Academic Editor: Agacik Zafer

Copyright © 2008 Khalik G. Guseinov and Ali S. Nazlipinar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. V. Beletskii, Studies of Motions of Celestial Bodies, Nauka, Moscow, Russia, 1972.
  2. N. N. Krasovskii, Theory of Control of Motion: Linear Systems, Nauka, Moscow, Russia, 1968. View at Zentralblatt MATH · View at MathSciNet
  3. V. I. Ukhobotov, One Dimensional Projection Method in Linear Differential Games with Integral Constraints, Chelyabinsk State University Press, Chelyabinsk, Russia, 2005.
  4. A. G. Chentsov, “Asymptotic attainability with perturbation of integral constraints in an abstract control problem—I,” Russian Mathematics, vol. 39, no. 2, pp. 57–68, 1995. View at Google Scholar · View at MathSciNet
  5. A. G. Chentsov, Asymptotic Attainability, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at Zentralblatt MATH
  6. R. Conti, Problemi di Controllo e di Controllo Ottimale, UTET, Torino, Italy, 1974.
  7. F. Gozzi and P. Loreti, “Regularity of the minimum time function and minimum energy problems: the linear case,” SIAM Journal on Control and Optimization, vol. 37, no. 4, pp. 1195–1221, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. W. Lou, “On the attainable sets of control systems with p-integrable controls,” Journal of Optimization Theory and Applications, vol. 123, no. 1, pp. 123–147, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Motta and C. Sartori, “Minimum time and minimum energy function for linear systems with controls in Lp for p1,” preprint, 200.
  10. A. N. Sirotin and A. M. Formalskii, “Reachability and controllability of discrete-time systems under control actions bounded in magnitude and norm,” Automation and Remote Control, vol. 64, no. 12, pp. 1844–1857, 2003. View at Publisher · View at Google Scholar
  11. Kh. G. Guseĭnov, A. A. Neznakhin, and V. N. Ushakov, “Approximate construction of reachable sets of control systems with integral constraints on the controls,” Journal of Applied Mathematics and Mechanics, vol. 63, no. 4, pp. 557–567, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Kh. G. Guseinov, O. Ozer, and E. Akyar, “On the continuity properties of the attainable sets of control systems with integral constraints on control,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 3, pp. 433–449, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Kh. G. Guseinov, O. Ozer, E. Akyar, and V. N. Ushakov, “The approximation of reachable sets of control systems with integral constraint on controls,” Nonlinear Differential Equations and Applications, vol. 14, no. 1-2, pp. 57–73, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. Motta and C. Sartori, “Minimum time with bounded energy, minimum energy with bounded time,” SIAM Journal on Control and Optimization, vol. 42, no. 3, pp. 789–809, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Kh. G. Guseinov and A. S. Nazlipinar, “On the continuity property of Lp balls and an application,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1347–1359, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Motta and F. Rampazzo, “Multivalued dynamics on a closed domain with absorbing boundary. Applications to optimal control problems with integral constraints,” Nonlinear Analysis: Theory, Methods & Applications, vol. 41, no. 5-6, pp. 631–647, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. B. T. Polyak, “Convexity of the reachable set of nonlinear systems under L2 bounded controls,” Tech. Rep. 02.2002/2003, Institut Mittag-Leffler, Djursholm, Sweden, 2003. View at Google Scholar
  18. P. Soravia, “Viscosity solutions and optimal control problems with integral constraints,” Systems & Control Letters, vol. 40, no. 5, pp. 325–335, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. F. Filippov, “On some questions in the theory of optimal regulation: existence of a solution of the problem of optimal regulation in the class of bounded measurable functions,” Vestnik Moskovskogo Universiteta, Seriya Matematika, Mekhanika, vol. 1959, no. 2, pp. 25–32, 1959. View at Google Scholar · View at MathSciNet
  20. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. View at Zentralblatt MATH · View at MathSciNet