Controllability and observability of linear impulsive differential algebraic system with Caputo fractional derivative

Document Type : Research Paper

Authors

1 Pakistan Institute of Engineering and Technology, Multan, Pakistan.

2 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.

3 Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Campus, Van, Turkey.

Abstract

Linear impulsive fractional differential-algebraic systems (LIFDAS) in a finite dimensional space are studied. We obtain the solution of LIFDAS. Using Gramian matrices, necessary and sufficient conditions for controllability and observability of time varying LIFDAS are established. We acquired the criterion for time-invariant LIFDAS in the form of rank conditions. The results are more generalized than the results that are obtained for various differential-algebraic systems without impulses. 

Keywords

References

  • [1]          R. Agarwal, S. Hristova, and D. O’Regan, Stability of solutions to impulsive Caputo fractional differential equa- tions, Electron. J. Differential Equations, (2016), 1-22.
  • [2]          E. Bazhlekova, Completely monotone functions and some classes of fractional evolution equations, Integral Trans- form Spec Funct., 26(9) (2015), 737–752.
  • [3]          T. Berger and T. Reis, Controllability of linear differential–algebraic systems a survey, In: Surveys in differential– algebraic equations I. Differential–algebraic equations forum, Berlin: Springer 2013, 1–61.
  • [4]          D. Boyadzhiev, H. Kiskinov, and A. Zahariev, Integral representation of solutions of fractional system with dis- tributed delays, Integral Transform Spec Funct., 29(9) (2018), 725-744.
  • [5]          S. L. Campbell, C. D. Meyer, and N. J. Rose, Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J Appl. Math., 31(3) (1976), 411-425.
  • [6]          E. Demirci and N. Ozlap, A method for solving differential equations of fractional order, Journal of Computational and Applied Mathematics, 236(11) (2012), 2754-2762.
  • [7]          M. Dodig, Matrix pencils completion problems, Linear Algebra and its Applications, 428(1) (2008), 259–304.
  • [8]          M. Feckan, Y. Zhou, and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(7) (2012), 3050-3060.
  • [9]          Z. Feng and N. Chen, On the Existence and Uniqueness of the Solution of Linear Fractional Differential-Algebraic System, Hindawi Publishing Corporation Mathematical Problems in Engineering (2016), Article ID 1674725.
  • [10]        J. M. Gracia, F. E. Velasco, Stability of invariant subspaces of regular matrix pencils, Linear Algebra and its applications, (1995), 219-226.
  • [11]        Z. H. Guan, J. Yao, and D. J. Hill, Robust H∞ control of singular impulsive systems with uncertain perturbation, IEEE Transactions on Circuits and Systems II: Express Briefs, 52(6) (2005), 293-298.
  • [12]        Z. H. Guan, T. H. Qian, and X. Yu, On controllability and observability for a class of impulsive systems, Systems and Control Letters, 47(3) (2002), 247-257.
  • [13]        T. L. Guo, Controllability and observability of impulsive fractional linear time invariant system, Comput. Math. Appl., 64(10) (2012), 3171-3182.
  • [14]        D. Guang-Ren, Analysis and Design of Descriptor Linear Systems, Springer, New York, 23 (2010).
  • [15]        N. A. Kablar, V. Kvrgic, and D. L. Debeljkovic, Singularly impulsive dynamical systems with time delay: Math- ematical model and stability, Transactions of Famena XXXVII-3, 37(3) (2013), 65-74.
  • [16]        N. A. Kablar, V. Kvrgic, and D. Ilic, Singularly Impulsive model of Genetic Regulatory Networks, 24th Chinese Control and Decision Conference (CCDC), Taiyuan, (2012), 113-117.
  • [17]        T. Kaczorek, Drazin inverse matrix method for fractional descriptor discrete time linear systems, Bulletin of the Polish Academy of Sciences Technical Sciences, 64 (2016), 395-399.
  • [18]        T. Kaczorek and K. Rogoeski, Fractional Linear Systems and Electrical Circuits,  Studies in Systems,  Decision  and control, Springer, Cham, 2015.
  • [19]        T. Kaczorek, Drazin inverse method for fractional descriptor continuous-time linear systems, Bulletin of the Polish Academy of Sciences Technical Sciences, 62(3) (2014), 409-412.
  • [20]        T. Kaczorek, Application of the Drazin inverse to the analysis of descriptor fractional discrete time linear systems with regular pencils, Int. J. Appl. Math. Comput. Sci., 23(1) (2013), 29-33 .
  • [21]        T. Kaczorek, Selected Problems of Fractional Systems Theory, LNCIS Vol. 411. Springer, Heidelberg, 2011.
  • [22]        F. Kusters, Switch Observability for Differential-Algebraic Systems.: Analysis, Observer Design and Application  to Power Networks, 2018. Thesis (Ph.D.)
  • [23]        N. S. Nise, Control Systems Engineering, 7th edition, John Wiley and Sons, Inc., 2014.
  • [24]        K. Ogata, Modern Control Engineering, Pearson, 5th edition, 2011.
  • [25]        K. B. Oldham and J. Spanier, The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering, Vol. 111. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.
  • [26]        I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [27]        W. J. Rugh, Linear System Theory, Prentice Hall Information and System Sciences Series, Prentice Hall, Inc., Englewood Cliffs, NJ, 1993.
  • [28]        V. Slynko and C. Tun¸c, Stability of abstract linear switched impulsive differential equations, Automatica J. IFAC 107, (2019), 433–441.
  • [29]        V. Slyn’ko and C. Tun¸c, Sufficient  conditions  for  stability  of  periodic  linear  impulse  delay  systems, Automation and Remote Control, 79(11) (2018), 1739–1754.
  • [30]        V. Slyn’ko and C. Tun¸c, Global asymptotic stability of nonlinear periodic impulsive equations, Miskolc Mathemat- ical Notes, 19(1) (2018), 595-610.
  • [31]        V.    Slyn’ko,    C.    Tun¸c,    and    V.O.    Bivziuk,    Robust   stabilization   of   nonlinear   nonautonomous   control systems with periodic linear approximation, IMA J. Math. Control Inform, 38(1) (2021), 125-142, https://doi.org/10.1093/imamci/dnaa003.
  • [32]        V. Slyn’ko, C. Tun¸c, and S. Erdur, Sufficient conditions of interval stability of a class of linear impulsive systems with a delay, Journal of Computer and Systems Sciences International, 59(1) (2020), 8–18.
  • [33]        E.  Tun¸c  and  O.  Tun¸c,  On  the  oscillation  of  a  class  of  damped  fractional  differential  equations,  Miskolc  Math. Notes, 17(1) (2016), 647656.
  • [34]        O. Tun¸c, On the qualitative analyses of integro-differential equations with constant time lag, Appl. Math. Inf. Sci, 14(1) (2020), 57-63.
  • [35]        C. Tun¸c, O. Tun¸c, A note on the stability and boundedness of solutions to non-linear differential systems of second order, Journal of the Association of Arab Universities for Basic and Applied Sciences, 24 (2017), 169-175.
  • [36]        I. Tejado, D. Valerio, and N. Valerio, Fractional calculus in economic growth modelling: the Spanish and Por- tuguese cases., Int. J. Dyn. Control, 5(1) (2017), 208–222.
  • [37]        L. A. Vlasenko and N. A. Perestyuk, On the solvability of differential-algebraic equations with impulse action, (Russian) Ukran. Mat. Zh., 57(4) (2005), 458–468; translation in Ukrainian Math. 57(4) (2005), 551–564
  • [38]        W. Wulling, The Drazin inverse of a singular, unreduced tridiagonal matrix, Linear Algebra and its Applications, 439 (10) (2013), 2736-2745.
  • [39]        A. Younas, I. Javaid, and A. Zehra, On controllability and observability of fractional continuous time linear systems with regular pencils, Bulletin of the Polish Academy of Sciences Technical Sciences, 65(3) (2017), 297–304.
  • [40]        X.F. Zhou, S. Liu, and W. Jiang, Complete controllability of impulsive  fractional linear time invarient systems  with delay, Abst. Appl. Anal., (2013), ID 374938.
Computational Methods for Differential Equations
Volume 10, Issue 1
January 2022
Pages 200-214

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History

  • Receive Date: 20 April 2020
  • Revise Date: 07 November 2020
  • Accept Date: 06 December 2020

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