Opuscula Math. 45, no. 2 (2025), 145-177
https://doi.org/10.7494/OpMath.2025.45.2.145

 
Opuscula Mathematica

Representation of solutions of linear discrete systems with constant coefficients and with delays

Josef Diblík

Abstract. The paper surveys the results achieved in representing solutions of linear non-homogeneous discrete systems with constant coefficients and with delays and their fractional variants by using special matrices called discrete delayed-type matrices. These are used to express solutions of initial problems in a closed and often simple form. Then, results are briefly discussed achieved by such representations of solutions in stability, controllability and other fields. In addition, a similar topic is dealt with concerning linear non-homogeneous differential equations with delays and their variants. Moreover, some comments are given to this parallel direction pointing some important moments in the developing this theory. An outline of future perspectives in this direction is discussed as well.

Keywords: discrete linear system, constants coefficients, delay, discrete matrix functions, representation of solutions, commutative matrices, non-commutative matrices, controllability.

Mathematics Subject Classification: 39A06, 39A12, 39A27, 39A30.

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  1. R.P. Agarwal, Difference Equations and Inequalities, 2nd ed., Marcel Dekker, Inc., 2000. https://doi.org/10.1201/9781420027020
  2. M. Awadalla, N.I. Mahmudov, J. Alahmadi, A novel delayed discrete fractional Mittag-Leffler function: representation and stability of delayed fractional difference system, J. Appl. Math. Comput. 70 (2024), no. 2, 1571-1599. https://doi.org/10.1007/s12190-024-02012-8
  3. N.V. Azbelev, P.M. Simonov, Stability of Differential Equations with Aftereffect, Stability and Control: Theory, Methods and Applications 20, Taylor & Francis, London, 2002. https://doi.org/10.1201/9781482264807
  4. N. Azbelev, V. Maksimov, L. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations, Advanced Series in Mathematical Science and Engineering, World Federation Publishers Company, Atlanta, GA, 1995. https://doi.org/10.1155/9789775945495
  5. A. Boichuk, J. Diblík, D. Khusainov, M.Růžičková, Fredholm's boundary-value problems for differential systems with a single delay, Nonlinear Anal. 42 (2010), no. 5, 2251-2258. https://doi.org/10.1016/j.na.2009.10.025
  6. X. Cao, J. Wang, Finite-time stability of a class of oscillating systems with two delays, Math. Methods Appl. Sci. 41 (2018), no. 13, 4943-4954. https://doi.org/10.1002/mma.4943
  7. Y. Chen, Representation of solutions and finite-time stability for fractional delay oscillation difference equations, Math. Methods Appl. Sci. 47 (2024), no. 6, 3997-4013. https://doi.org/10.1002/mma.9799
  8. P.A. Cojuhari, A.M. Gomilko, On the characterization of scalar type spectral operators, Studia Math. 184 (2008), no. 2, 121-132. https://doi.org/10.4064/sm184-2-2
  9. J. Diblík, Relative and trajectory controllability of linear discrete systems with constant coefficients and a single delay, IEEE Trans. Automat. Control 64 (2019), 2158-2165. https://doi.org/10.1109/tac.2018.2866453
  10. J. Diblík, Bounded solutions to systems of fractional discrete equations, Adv. Nonlinear Anal. 11 (2022), no. 1, 1614-1630. https://doi.org/10.1515/anona-2022-0260
  11. J. Diblík, K. Mencáková, Solving a higher-order linear discrete systems, [in:] Mathematics, Information Technologies and Applied Sciences 2017, post-conference proceedings of extended versions of selected papers, Brno, University of Defence, 2017, pp. 77-91.
  12. J. Diblík, K. Mencáková, Solving a higher-order linear discrete equation, [in:] Proceedings, 16th Conference on Applied Mathematics Aplimat 2017, Bratislava, 2017, pp. 445-453.
  13. J. Diblík, K. Mencáková, A note on relative controllability of higher-order linear delayed discrete systems, IEEE Trans. Automat. Control 65 (2020), 5472-5479. https://doi.org/10.1109/tac.2020.2976298
  14. J. Diblík, K. Mencáková, Representation of solutions to delayed linear discrete systems with constant coefficients and with second-order differences, Appl. Math. Lett. 105 (2020), 106309. https://doi.org/10.1016/j.aml.2020.106309
  15. J. Diblík, B. Morávková, Discrete matrix delayed exponential for two delays and its property, Adv. Difference Equ. 2013 (2013), Article no. 139. https://doi.org/10.1186/1687-1847-2013-139
  16. J. Diblík, B. Morávková, Representation of the solutions of linear discrete systems with constant coefficients and two delays, Abstr. Appl. Anal. 2014 (2014), Art. ID 320476. https://doi.org/10.1155/2014/320476
  17. J. Diblík, D.Ya. Khusainov, Representation of solutions of discrete delayed system \(x(k+1)=Ax(k)+Bx(k-m)+f(k)\) with commutative matrices, J. Math. Anal. Appl. 318 (2006), no. 1, 63-76. https://doi.org/10.1016/j.jmaa.2005.05.021
  18. J. Diblík, D.Ya. Khusainov, Representation of solutions of linear discrete systems with constant coefficients and pure delay, Adv. Difference Equ. 2006 (2006), 1-13. https://doi.org/10.1155/ade/2006/80825
  19. J. Diblík, D.Ya. Khusainov, M. Růžičková, Controllability of linear discrete systems with constant coefficients and pure delay, SIAM J. Control Optim. 47 (2008), 1140-1149. https://doi.org/10.1137/070689085
  20. J. Diblík, D.Ya. Khusainov, J. Lukáčová, M. Růžičková, Control of oscillating systems with a single delay, Adv. Difference Equ. 2010, Art. ID 108218. https://doi.org/10.1186/1687-1847-2010-108218
  21. J. Diblík, M. Fečkan, M. Pospíšil, On the new control functions for linear discrete delay systems, SIAM J. Control Optim. 52 (2014), 1745-1760. https://doi.org/10.1137/140953654
  22. F. Du, B. Jia, Finite time stability of fractional delay difference systems: A discrete delayed Mittag-Leffler matrix function approach, Chaos Solitons Fractals 141 (2020), 110430. https://doi.org/10.1016/j.chaos.2020.110430
  23. F. Du, J.-G. Lu, Exploring a new discrete delayed Mittag-Leffler matrix function to investigate finite-time stability of Riemann-Liouville fractional-order delay difference systems, Math. Methods Appl. Sci. 45 (2022), no. 16, 9856-9878. https://doi.org/10.1002/mma.8342
  24. S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer, 2005.
  25. A. Elshenhab, X.T. Wang, Representation of solutions of delayed linear discrete systems with permutable or nonpermutable matrices and second-order differences, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 2, Paper no. 58. https://doi.org/10.1007/s13398-021-01204-2
  26. C. Goodrich, A. Peterson, Discrete Fractional Calculus, Springer, New York, 2015.
  27. X. Jin, J. Wang, Iterative learning control for linear discrete delayed systems with non-permutable matrices, Bull. Iranian Math. Soc. 28 (2022), no. 4, 1553-1574. https://doi.org/10.1007/s41980-021-00593-9
  28. X. Jin, J. Wang, D. Shen, Convergence analysis for iterative learning control of impulsive linear discrete delay systems, J. Difference Equ. Appl. 27 (2021), no. 5, 739-762. https://doi.org/10.1080/10236198.2021.1938562
  29. X. Jin, J. Wang, D. Shen, Representation and stability of solutions for impulsive discrete delay systems with linear parts defined by non-permutable matrices, Qual. Theory Dyn. Syst. 21 (2022), no. 4, Paper 152. https://doi.org/10.1007/s12346-022-00685-9
  30. X. Jin, M. Fečkan, J. Wang, Relative controllability of impulsive linear discrete delay systems, Qual. Theory Dyn. Syst. 22 (2023), Paper no. 133. https://doi.org/10.1007/s12346-023-00831-x
  31. D.Ya. Khusainov, G.V. Shuklin, Linear autonomous time-delay system with permutation matrices solving, Stud. Univ. Žilina Math. Ser. 17 (2023), 101-108.
  32. D.Ya. Khusainov, J. Diblík, M. Růžičková, J. Lukáčová, A representation of the solution of the Cauchy problem for an oscillatory system with pure delay Nonlinear Oscil. 11 (2008), no. 2, 276-285. https://doi.org/10.1007/s11072-008-0030-8
  33. D. Khusainov, J. Diblík, M. Růžičková, Linear Dynamical Systems with Aftereffect, Representation of Solutions, Stability, Control, Stabilization, Kiev National University named after Taras Shevchenko, Kiev, 2015 [in Russian].
  34. V. Lakshmikantham, D. Trigiante, Theory of Difference Equations, 2nd ed., Marcel Dekker, Inc., 2002. https://doi.org/10.1201/9780203910290
  35. M. Li, J. Wang, Existence results and Ulam type stability for conformable fractional oscillating system with pure delay, Chaos Solitons Fractals 161 (2022), 112317. https://doi.org/10.1016/j.chaos.2022.112317
  36. Y. Liang, Y. Shi, Z. Fan, Exact solutions and Hyers-Ulam stability of fractional equations with double delays, Fract. Calc. Appl. Anal. 26 (2023), no. 1, 439-460. https://doi.org/10.1007/s13540-022-00122-3
  37. C. Liang, J. Wang, D. O'Regan, Controllability of nonlinear delay oscillating systems, Electron. J. Qual. Theory Differ. Equ. 2017, Paper no. 47. https://doi.org/10.14232/ejqtde.2017.1.47
  38. C. Liang, J. Wang, M. Fečkan, A study on ILC for linear discrete systems with single delay, J. Difference Equ. Appl. 24 (2018), no. 3, 358-374.
  39. C. Liang, J. Wang, D. Shen, Iterative learning control for linear discrete delay systems via discrete matrix delayed exponential function approach, J. Difference Equ. Appl. 24 (2018), 1756-1776. https://doi.org/10.1080/10236198.2018.1529762
  40. C. Liang, J. Wang, D. O'Regan, Representation of a solution for a fractional linear system with pure delay, Appl. Math. Lett. 77 (2018), 72-78. https://doi.org/10.1016/j.aml.2017.09.015
  41. H. Luo, J. Wang, D. Shen, Learning ability analysis for linear discrete delay systems with iteration-varying trial length, Chaos, Solitons and Fractals 171 (2023), 113428. https://doi.org/10.1016/j.chaos.2023.113428
  42. N.I. Mahmudov, Representation of solutions of discrete linear delay systems with non permutable matrices, Appl. Math. Lett. 85 (2018), 8-14. https://doi.org/10.1016/j.aml.2018.05.015
  43. N.I. Mahmudov, A novel fractional delayed matrix cosine and sine, Appl. Math. Lett. 92 (2019), 41-48. https://doi.org/10.1016/j.aml.2019.01.001
  44. N.I. Mahmudov, Delayed perturbation of Mittag-Leffler functions and their applications to fractional linear delay differential equations, Math. Methods Appl. Sci. 42 (2019), 5489-5497. https://doi.org/10.1002/mma.5446
  45. N.I. Mahmudov, Delayed linear difference equations: the method of \(Z\)-transform, Electron. J. Qual. Theory Differ. Equ. (2020), Paper no. 53. https://doi.org/10.14232/ejqtde.2020.1.53
  46. N.I. Mahmudov, Analytical solution of the fractional linear time-delay systems and their Ulam-Hyers stability, J. Appl. Math. (2022), Art. ID 2661343. https://doi.org/10.1155/2022/2661343
  47. N.I. Mahmudov, Multi-delayed perturbation of Mittag-Leffler type matrix functions, J. Math. Anal. Appl. 505 (2022), 125589. https://doi.org/10.1016/j.jmaa.2021.125589
  48. N.I. Mahmudov, Multiple delayed linear difference equations with non-permutable matrix coefficients: the method of \(Z\)-transform, Montes Taurus J. Pure and Appl. Math. 6 (2024), 138-146.
  49. M. Medved', M. Pospíšil, Representation and stability of solutions of systems of difference equations with multiple delays and linear parts defined by pairwise permutable matrices, Commun. Appl. Anal. 17 (2013), no. 1, 21-45.
  50. M. Medved', L. Škripková, Sufficient conditions for the exponential stability of delay difference equations with linear parts defined by permutable matrices, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), no. 22. https://doi.org/10.14232/ejqtde.2012.1.22
  51. K. Mencáková, J. Diblík, Relative controllability of a linear system of discrete equations with single delay, AIP Conf. Proc. 2293 (2020), 340009. https://doi.org/10.1063/5.0026620
  52. B. Morávková, Representation of solutions of linear discrete systems with delay, Ph.D. Thesis, University of Technology, Brno, Czech Republic, 2024.
  53. B. Morávková, J. Diblík, Solutions of linear discrete systems with a single delay and impulses, AIP Conf. Proc. 2849 (2023), 370003. https://doi.org/10.1063/5.0162562
  54. M. Pospíšil, Representation and stability of solutions of systems of functional differential equations with multiple delays, Electron. J. Qual. Theory Differ. Equ. (2012), no. 54. https://doi.org/10.14232/ejqtde.2012.1.54
  55. M. Pospíšil, Relative controllability of delayed difference equations to multiple consecutive states, AIP Conf. Proc. 1863 (2017), 480002. https://doi.org/10.1063/1.4992638
  56. M. Pospíšil, Representation of solutions of delayed difference equations with linear parts given by pairwise permutable matrices via \(\mathcal{Z}\)-transform, Appl. Math. Comput. 294 (2017), 180-194. https://doi.org/10.1016/j.amc.2016.09.019
  57. M. Pospíšil, J. Diblík, M. Fečkan, Observability of difference equations with a delay, AIP Conf. Proc. 1558 (2013), 478-481. https://doi.org/10.1063/1.4825531
  58. M. Pospíšil, J. Diblík, M. Fečkan, On relative controllability of delayed difference equations with multiple control functions, AIP Conf. Proc. 1648 (2015), 130001. https://doi.org/10.1063/1.4912420
  59. T. Sathiyaraj, J. Wang, Controllability and stability of non-instantaneous impulsive stochastic multiple delays system, J. Optim. Theory Appl. 201 (2024), no. 3, 995-1025. https://doi.org/10.1007/s10957-024-02430-5
  60. Z. Svoboda, Asymptotic unboundedness of the norms of delayed matrix sine and cosine, Electron. J. Qual. Theory Differ. Equ. (2017), no. 89. https://doi.org/10.14232/ejqtde.2017.1.89
  61. J. Wang, Z. Luo, M. Fečkan, Relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices, Eur. J. Control 38 (2017), 39-46. https://doi.org/10.1016/j.ejcon.2017.08.002
  62. J. Wang, Z. Luo, D. Shen, Iterative learning control for linear delay systems with deterministic and random impulses, J. Franklin Inst. 355 (2018), no. 5, 2473-2497. https://doi.org/10.1016/j.jfranklin.2018.01.013
  63. J. Wang, M. Fečkan, M. Li, Stability and Controls Analysis for Delay Systems, Academic Press, 2022.
  64. M. Yang, M. Fečkan, J. Wang, Relative controllability for delayed linear discrete system with second-order differences, Qual. Theory Dyn. Syst. 21 (2022), Article no. 113. https://doi.org/10.1007/s12346-022-00645-3
  65. M. Yang, M. Fečkan, J. Wang, Solution to delayed linear discrete system with constant coefficients and second-order differences and application to iterative learning control, Int. J. Adapt. Control Signal Process. 38 (2024), 677-695. https://doi.org/10.1002/acs.3722
  66. M. Yang, M. Fečkan, J. Wang, Ulam's type stability of delayed discrete system with second-order differences, Qual. Theory Dyn. Syst. 23 (2024), Article no. 11. https://doi.org/10.1007/s12346-023-00868-y
  67. E. Zalot, Spectral resolutions for non-self-adjoint block convolution operators, Opuscula Math. 42 (2022), no. 3, 459-487. https://doi.org/10.7494/opmath.2022.42.3.459
  68. E. Zalot, W. Majdak, Spectral representations for a class of banded Jacobi-type matrices, Opuscula Math. 34 (2014), no. 4, 871-887. https://doi.org/10.7494/opmath.2014.34.4.871
  69. A. Zhou, Exponential stability and relative controllability of nonsingular conformable delay systems, Axioms 12 (2023), 994. https://doi.org/10.3390/axioms12100994
  • Josef Diblík
  • ORCID iD https://orcid.org/0000-0001-5009-316X
  • Brno University of Technology, Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno, Czech Republic
  • Brno University of Technology, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno, Czech Republic
  • Brno University of Technology, Central European Institute of Technology, Division of Cybernetics and Robotics, Brno, Czech Republic
  • Communicated by P.A. Cojuhari.
  • Received: 2024-11-21.
  • Accepted: 2024-12-23.
  • Published online: 2025-03-10.
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Cite this article as:
Josef Diblík, Representation of solutions of linear discrete systems with constant coefficients and with delays, Opuscula Math. 45, no. 2 (2025), 145-177, https://doi.org/10.7494/OpMath.2025.45.2.145

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