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Optimal Solutions for Non-consistent Singular Linear Systems of Fractional Nabla Difference Equations

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Abstract

In this article, we study optimal solutions for a class of non-consistent singular linear systems of fractional nabla difference equations whose coefficients are constant matrices. We take into consideration the cases that the matrices are square with the leading coefficient singular, non-square and square with a matrix pencil which has an identically zero determinant. Then, first we study the system with given non-consistent initial conditions and provide optimal solutions. Furthermore, we consider the system with boundary conditions and provide optimal solutions for two cases, when the boundary value problem is non-consistent and when it has infinite solutions. Finally, we study the Kalman filter for singular non-homogeneous linear control systems of fractional nabla difference equations. Numerical examples are given to justify our theory.

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References

  1. R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109(3), 973–1033 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. K. Ahrendt, L. Castle, M. Holm, K. Yochman, Laplace transforms for the nabla difference operator and a fractional variation of parameters formula. Commun. Appl. Anal. 16(3), 317 (2012)

    MATH  MathSciNet  Google Scholar 

  3. R. Almeida, A.B. Malinowska, D.F.M. Torres, Fractional Euler Lagrange Differential Equations via Caputo Derivatives, Fractional Dynamics and Control (Springer, Berlin, 2012)

    Google Scholar 

  4. F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. F.M. Atici, P.W. Eloe, Modeling with fractional difference equations. J. Math. Anal. Appl. 369(1), 1–9 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. F.M. Atici, P.W. Eloe, Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 41(2), 353–370 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Baleanu, O.G. Mustafa, OG, On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59(5), 1835–1841 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Baleanu, A. Babakhani, Employing of some basic theory for class of fractional differential equations. Adv. Differ. Equ. 2011, 296353 (2011)

    MathSciNet  Google Scholar 

  9. D. Baleanu, K. Diethelm, E. Scalas, Fractional Calculus: Models and Numerical Methods (World Scientific, Hackensack, 2012)

    Google Scholar 

  10. R.G. Brown, P.Y.C. Hwang, Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises and Solutions (Wiley, New York, 1997)

    MATH  Google Scholar 

  11. S.L. Campbell, Singular Systems of Differential Equations, vol. 1 (Pitman, San Francisco, 1980), vol. 2 (1982)

  12. L. Dai, Singular Control Systems. Lecture Notes in Control and Information Sciences, ed. by M. Thoma, A. Wyner (1988)

  13. I. Dassios, On non-homogeneous linear generalized linear discrete time systems. Circuits Syst. Signal Process. 31(5), 1699–1712 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. I. Dassios, On robust stability of autonomous singular linear matrix difference equations. Appl. Math. Comput. 218(12), 6912–6920 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. I. Dassios, G. Kalogeropoulos, On a non-homogeneous singular linear discrete time system with a singular matrix pencil. Circuits Syst. Signal Process. 32(4), 1615–1635 (2013)

    Article  MathSciNet  Google Scholar 

  16. I.K. Dassios, D. Baleanu, On a singular system of fractional nabla difference equations with boundary conditions. Bound. Value Probl. 2013, 148 (2013)

    Article  MathSciNet  Google Scholar 

  17. I. Dassios, G. Kalogeropoulos, On the relation between consistent and non consistent initial conditions of singular discrete time systems. Dyn. Contin. Discret. impuls. Syst. Ser. A 20(4a), 447–458 (2013)

    MATH  MathSciNet  Google Scholar 

  18. I. Dassios, D. Baleanu, G. Kalogeropoulos, On non-homogeneous singular systems of fractional nabla difference equations. Appl. Math. Comput. 227, 112–131 (2014)

    Article  MathSciNet  Google Scholar 

  19. B. Datta, Numerical Linear Algebra and Applications (Siam, Philadelphia, 2010)

    Book  MATH  Google Scholar 

  20. A. Dzieliski, D. Sierociuk, Stability of discrete fractional order state-space systems. J. Vib. Control 14(9–10), 1543–1556 (2008)

    Article  MathSciNet  Google Scholar 

  21. A. Dzieliski, D. Sierociuk, New Trends in Nanotechnology and Fractional Calculus Applications, Fractional order model of beam heating process and its experimental verification (Springer, New York, 2010)

    Google Scholar 

  22. A. Dzieliski, W. Malesza, Point to point control of fractional differential linear control systems. Adv. Differ. Equ. 13, 17 (2011)

    Google Scholar 

  23. A. Dzieliski, G. Sarwas, D. Sierociuk, Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models. Adv. Differ. Equ. 11, 15 (2011)

    Google Scholar 

  24. R.F. Gantmacher, The Theory of Matrices I, II (New York, Chelsea, 1959)

    Google Scholar 

  25. W.G. Glockle, T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68(1), 46–53 (1995)

    Article  Google Scholar 

  26. G. Golub, C.V. Loan, Matrix Computations, vol. 3 (JHU Press, London, 2012)

    Google Scholar 

  27. E. Grispos, S. Giotopoulos, G. Kalogeropoulos, On generalised linear discrete-time regular delay systems. J. Inst. Math. Comput. Sci. Math. Ser. 13(2), 179–187 (2000)

    MATH  MathSciNet  Google Scholar 

  28. E. Grispos, G. Kalogeropoulos, I. Stratis, On generalised linear discrete-time singular delay systems. J. Math. Anal. Appl. 245(2), 430–446 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. J. Hein, Z. McCarthy, N. Gaswick, B. McKain, K. Speer, Laplace transforms for the nabla difference operator. Panam. Math. J. 21(3), 79–97 (2011)

    MATH  MathSciNet  Google Scholar 

  30. R. Hilfe, Applications of Fractional Calculus in Physics (World Scientific, River Edge, 2000), p. 463

    Book  Google Scholar 

  31. T. Kaczorek, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, Positive fractional linear electrical circuits (International Society for Optics and Photonics, San Francisco, 2013)

    Google Scholar 

  32. T. Kaczorek, Singular fractional continuous-time and discrete-time linear systems. Acta Mech. Autom. 7(1), 26–33 (2013)

    Google Scholar 

  33. R.E. Kalman, A new approach to linear filtering and prediction problems. ASME J. Basic Eng. Ser. D 82, 3545 (1960)

    Google Scholar 

  34. G.I. Kalogeropoulos, Matrix pencils and linear systems, Ph.D Thesis, City University, London (1985)

  35. G. Kalogeropoulos, I.G. Stratis, On generalized linear regular delay systems. J. Math. Anal. Appl. 237(2), 505–514 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  36. J. Klamka, Controllability of dynamical systems. Mat. Stosow. 50(9), 57–75 (2008)

    MathSciNet  Google Scholar 

  37. J. Klamka, J. Wyrwa, Controllability of second-order in finite-dimensional systems. Syst. Control Lett. 57(5), 386–391 (2008)

    Article  MATH  Google Scholar 

  38. J. Klamka, Monograph New Trends in Nanotechnology and Fractional Calculus, Controllability and minimum energy control problem of fractional discrete-time systems (Springer, New York, 2010)

    Google Scholar 

  39. T.D. Lee, Can time be a discrete dynamical variable? Phys. Lett. B 122(3–4), 217–220 (1983)

    Article  Google Scholar 

  40. A.B. Malinowska, D.F.M. Torres, Introduction to the Fractional Calculus of Variations, vol. 16 (Imperial College Press, London, 2012), p. 275

    Book  MATH  Google Scholar 

  41. M. Mitrouli, G. Kalogeropoulos, A compound matrix algorithm for the computation of the Smith form of a polynomial matrix. Numer. Algorithms 7(2–4), 145–159 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  42. A. Nagai, Discrete Mittag–Leffler function and its applications. Publ. Res. Inst. Math. Sci. Kyoto Univ. 1302, 120 (2003)

    Google Scholar 

  43. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering (Academic Press, San Diego, 1999)

    Google Scholar 

  44. A.P. Schinnar, The Leontief dynamic generalized inverse. Q. J. Econ. 92(4), 641–652 (1978)

    Article  MATH  Google Scholar 

  45. J. Schutter, J. Geeter, T. Lefebvre, H. Bruynickx. Kalman lters: A tutorial, (1999)

  46. D. Sierociuk, A. Dzieliski, Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation. Int. J. Appl. Math. Comput. Sci. 16(1), 129–140 (2006)

    MathSciNet  Google Scholar 

  47. D. Sierociuk, G. Dzieliski, G. Sarwas, I. Petras, I. Podlubny, T. Skovranek, Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371(1990), 10 (2013)

    Article  Google Scholar 

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Acknowledgments

The author would like to thank the anonymous referees for their valuable suggestions. This work was partly funded by EPSRC Grant EP/I017127/1 and by Science Foundation Ireland (award 09/SRC/E1780).

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Authors and Affiliations

  1. MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland

    Ioannis K. Dassios

  2. School of Mathematics and Maxwell Institute, The University of Edinburgh, Edinburgh, UK

    Ioannis K. Dassios

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Dassios, I.K. Optimal Solutions for Non-consistent Singular Linear Systems of Fractional Nabla Difference Equations. Circuits Syst Signal Process 34, 1769–1797 (2015). https://doi.org/10.1007/s00034-014-9930-2

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  • DOI: https://doi.org/10.1007/s00034-014-9930-2

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