`Abstract and Applied AnalysisVolume 2011, Article ID 161246, 25 pageshttp://dx.doi.org/10.1155/2011/161246`
Research Article

## Positivity and Stability of the Solutions of Caputo Fractional Linear Time-Invariant Systems of Any Order with Internal Point Delays

Institute for Research and Development of Processes, Faculty of Science and Technology, University of Basque Country, Campus of Leioa, Aptdo. 544, 48080 Bilbao, Spain

Received 21 September 2010; Revised 15 November 2010; Accepted 11 January 2011

Copyright © 2011 M. De la Sen. 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.

1. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
2. Z. Odibat, “Approximations of fractional integrals and Caputo fractional derivatives,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 527–533, 2006.
3. Y. Luchko and R. Gorenflo, “An operational method for solving fractional differential equations with the Caputo derivatives,” Acta Mathematica Vietnamica, vol. 24, no. 2, pp. 207–233, 1999.
4. Y. F. Luchko and H. M. Srivastava, “The exact solution of certain differential equations of fractional order by using operational calculus,” Computers & Mathematics with Applications, vol. 29, no. 8, pp. 73–85, 1995.
5. R. C. Soni and D. Singh, “Certain fractional derivative formulae involving the product of a general class of polynomials and the multivariable H-function,” Proceedings of the Indian Academy of Sciences (Mathematical Sciences), vol. 112, no. 4, pp. 551–562, 2002.
6. R. K. Raina, “A note on the fractional derivatives of a general system of polynomials,” Indian Journal of Pure and Applied Mathematics, vol. 16, no. 7, pp. 770–774, 1985.
7. A. Babakhani and V. Daftardar-Gejji, “On calculus of local fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 66–79, 2002.
8. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009.
9. B. Baeumer, M. M. Meerschaert, and J. Mortensen, “Space-time fractional derivative operators,” Proceedings of the American Mathematical Society, vol. 133, no. 8, pp. 2273–2282, 2005.
10. B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 480–487, 2010.
11. F. Riewe, “Mechanics with fractional derivatives,” Physical Review E, vol. 55, no. 3, pp. 3581–3592, 1997.
12. A. M. A. El- Sayed and M. Gaber, “On the finite Caputo and finite Riesz derivatives,” Electronic Journal of Theoretic Physics, vol. 3, no. 12, pp. 81–95, 2006.
13. R. Almeida, A. B. Malinowska, and D. F. M. Torres, “A fractional calculus of variations for multiple integrals with application to vibrating string,” Journal of Mathematical Physics, vol. 51, no. 3, pp. 1–12, 2010.
14. I. Schäfer and S. Kempfle, “Impulse responses of fractional damped systems,” Nonlinear Dynamics, vol. 38, no. 1–4, pp. 61–68, 2004.
15. F. J. Molz III, G. J. Fix III, and S. Lu, “A physical interpretation for the fractional derivative in Levy diffusion,” Applied Mathematics Letters, vol. 15, no. 7, pp. 907–911, 2002.
16. M. Ilic, I. W. Turner, F. Liu, and V. Anh, “Analytical and numerical solutions of a one-dimensional fractional-in-space diffusion equation in a composite medium,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2248–2262, 2010.
17. M. D. Ortigueira, “On the initial conditions in continuous-time fractional linear systems,” Signal Processing, vol. 83, no. 11, pp. 2301–2309, 2003.
18. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M. Fernández-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Applied Optics, vol. 36, no. 29, pp. 7461–7464, 1997.
19. E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuous-time finance,” Physica A, vol. 284, no. 1–4, pp. 376–384, 2000.
20. S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, Germany, 2008.
21. J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010.
22. J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machad, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
23. L. A. Zadeh and C. A. Desoer, Linear Systems Theory: The State Space Approach, McGraw- Hill, New York, NY, USA, 1963.
24. M. Delasen, “Application of the non-periodic sampling to the identifiability and model matching problems in dynamic systems,” International Journal of Systems Science, vol. 14, no. 4, pp. 367–383, 1983.
25. M. De la Sen and N. Luo, “On the uniform exponential stability of a wide class of linear time-delay systems,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 456–476, 2004.
26. M. De la Sen, “Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 621–650, 2006.
27. J. Chen, D. M. Xu, and B. Shafai, “On sufficient conditions for stability independent of delay,” IEEE Transactions on Automatic Control, vol. 40, no. 9, pp. 1675–1680, 1995.
28. M. De la Sen, “On the reachability and controllability of positive linear time-invariant dynamic systems with internal and external incommensurate point delays,” Rocky Mountain Journal of Mathematics, vol. 40, no. 1, pp. 177–207, 2010.
29. M. De la Sen, “A method for general design of positive real functions,” IEEE Transactions on Circuits and Systems I, vol. 45, no. 7, pp. 764–769, 1998.
30. M. De la Sen, “Preserving positive realness through discretization,” Positivity, vol. 6, no. 1, pp. 31–45, 2002.
31. M. De la Sen, “On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 382–401, 2007.
32. M. De La Sen and S. Alonso-Quesada, “A simple vaccination control strategy for the SEIR epidemic model,” in Proceedings of the 5th IEEE International Conference on Management of Innovation and Technology (ICMIT '10), pp. 1037–1044, June 2010.
33. M. De La Sen and S. Alonso-Quesada, “On vaccination control tools for a general SEIR-epidemic model,” in Proceedings of the 18th Mediterranean Conference on Control and Automation (MED '10), pp. 1322–1328, June 2010.
34. R. P. Agarwal, Y. Sun, and P. J. Y. Wong, “Existence of positive periodic solutions of periodic boundary value problem for second order ordinary differential equations,” Acta Mathematica Hungarica, vol. 129, no. 1, pp. 166–181, 2010.
35. M. De la Sen, “On Chebyshev systems and non-uniform sampling related to Caputo fractional dynamic time-invariant systems,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 846590, 24 pages, 2010.