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Journal of Control Science and Engineering
Volume 2016, Article ID 4873083, 12 pages
http://dx.doi.org/10.1155/2016/4873083
Research Article

Optimal Control Problem Investigation for Linear Time-Invariant Systems of Fractional Order with Lumped Parameters Described by Equations with Riemann-Liouville Derivative

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Profsoyuznaya Street 65, Moscow 117997, Russia

Received 29 November 2015; Accepted 3 May 2016

Academic Editor: Francisco Gordillo

Copyright © 2016 V. A. Kubyshkin and S. S. Postnov. 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. A. G. Butkovskii, S. S. Postnov, and E. A. Postnova, “Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: modeling and hardware implementation,” Automation and Remote Control, vol. 74, no. 5, pp. 725–749, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. R. Kamocki, “Pontryagin maximum principle for fractional ordinary optimal control problems,” Mathematical Methods in the Applied Sciences, vol. 37, no. 11, pp. 1668–1686, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. R. Kamocki and M. Majewski, “Fractional linear control systems with Caputo derivative and their optimization,” Optimal Control Applications and Methods, vol. 36, no. 6, pp. 953–967, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. G. Butkovskiy, Distributed Control Systems, American Elsevier, New York, NY, USA, 1969.
  5. V. A. Kubyshkin and S. S. Postnov, “Optimal control problem for a linear stationary fractional order system in the form of a problem of moments: problem setting and a study,” Automation and Remote Control, vol. 75, no. 5, pp. 805–817, 2014. View at Publisher · View at Google Scholar
  6. V. A. Kubyshkin and S. S. Postnov, “Optimal control problem for linear fractional-order systems,” in Proceedings of the International Conference on Fractional Differentiation and Its Applications (ICFDA '14), p. 6, IEEE, Catania, Italy, June 2014, Paper ID 3189701.
  7. V. A. Kubyshkin and S. S. Postnov, “Analysis of two optimal control problems for a fractional-order pendulum by the method of moments,” Automation and Remote Control, vol. 76, no. 7, pp. 1302–1314, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. V. A. Kubyshkin and S. S. Postnov, “The optimal control problem for linear systems of non-integer order with lumped and distributed parameters, Discontinuity,” Nonlinearity and Complexity, vol. 4, no. 4, pp. 429–443, 2015. View at Google Scholar
  9. A. A. Kilbas, H. M. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  10. R. Almeida and D. F. M. Torres, “Calculus of variations with fractional derivatives and fractional integrals,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1816–1820, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. V. E. Tarasov, Fractional Dynamics, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Publications, Mineola, NY, USA, 1999.
  14. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner, New York, NY, USA, 1965. View at MathSciNet
  15. M. G. Krein and A. A. Nudelman, “The Markov moment problem and extremal problems,” in Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development, vol. 50 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1977. View at Google Scholar
  16. A. A. Feldbaum, Optimal Control Systems, Academic Press, New York, NY, USA, 1965. View at MathSciNet
  17. A. G. Butkovskiy, Phase Portraits of Control Dynamical Systems, Kluwer Academic, Dordrecht, The Netherlands, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  18. C. Li, D. Qian, and Y. Chen, “On Riemann-Liouville and caputo derivatives,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 562494, 15 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y. Li, Y. Q. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function.
  21. L. F. Shampine, “Vectorized adaptive quadrature in Matlab,” Journal of Computational and Applied Mathematics, vol. 211, no. 2, pp. 131–140, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus