Representation and Stability of Solutions for Impulsive Discrete Delay Systems with Linear Parts Defined by Non-Permutable Matrices

• Xianghua Jin ^[1] ; JinRong Wang ^[3] ; Dong Shen ^[2]
  1. [1] Guizhou University

     Guizhou University

     China

     
  2. [2] Renmin University of China

     Renmin University of China

     China

     
  3. [3] Guizhou University & Huaqiao University

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • In this paper, we introduce a modified delayed perturbation of discrete matrix exponential for impulsive linear discrete delay systems with non-permutable matrices. Using the Z-transform method and its alternative method, we derive a clear representation of the solution, which covers both impulsive and non-impulsive cases in the existing literature. Moreover, using the representation of solution, discrete Gronwall’s inequality, and discrete Bihari’s inequality, we establish several results of exponential stability.

    Numerical examples are given to verify these results.

• Referencias bibliográficas
  • 1. Richards, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)
  • 2. Diblík, J., Khusainov, D. Ya.: Representation of solutions of linear discrete systems with constant coefficients and pure delay. Adv. Difference...
  • 3. Khusainov, D Ya., Shuklin, G.V.: Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Žilina 17, 101–108...
  • 4. Diblík, J., Khusainov, D Ya.: Representation of solutions of discrete delayed system x(k + 1) = Ax(k) + Bx(k − m) + f (k)...
  • 5. Mahmudov, N.I.: Representation of solutions of discrete linear delayed systems with non permutable matrices. Appl. Math. Lett. 85, 8–14...
  • 6. Diblík, J., Morávková, B.: Representation of the solutions of linear discrete systems with constant coefficients and two delays. Abstr....
  • 7. Mahmudov, N.I.: Delayed linear difference equations: the method of Z-transform. Electron. J. Qual. Theory Differ. Equ. 53, 1–12 (2020)
  • 8. Pospíšil, M.: Representation of solutions of delayed difference equations with linear parts given by pairwise permutable matrices via Z-transform....
  • 9. Diblík, J., Morávková, B.: Representation of solutions of linear discrete systems with constant coefficients, a single delay and with impulses....
  • 10. Diblík, J.,Morávková, B.: Representation of solutions of impulsive linear discrete systems with constant coefficients and a single delay....
  • 11. Diblík, J., Khusainov, D. Ya., Kukharenko, O., Morávková, B.: Delayed exponential functions and their application to representations of...
  • 12. Morávková, B.: Representation of solutions of linear discrete systems with delay. Ph. D. Thesis, Brno University of Technology, Brno,...
  • 13. Medved’, M., Pospíšil, M., Škripková, L.: Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear...
  • 14. Medved’, M., Pospíšil, M.: Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear...
  • 15. You, Z., Wang, J.: Stability of impulsive delay differential equations. J. App. Math. Comput. 56, 253–268 (2018)
  • 16. You, Z., Wang, J., O’Regan, D.: Asymptotic stability of solutions of impulsive multi-delay differential equations. Trans. Inst. Measure....
  • 17. You, Z., Wang, J.: On the exponential stability of nonlinear delay systems with impulses. IMA J. Math. Control Inform. 35, 773–803 (2018)
  • 18. Berezansky, L., Diblík, J., Švoboda, Z., Smarda, Z.: Simple tests for uniform exponential stability of a linear delayed vector differential...
  • 19. Medved’, M., Škripková, L.: Sufficient conditions for the exponential stability of delay difference equations with linear parts defined...
  • 20. Medved’, M., Pospíšil, M.: Representation and stability of solutions of systems of difference equations with multiple delays and linear...
  • 21. Diblík, J., Khusainov, D.. Ya.., Baštinec, J., Sirenko, A.S.: Exponential stability of linear discrete systems with constant coefficients...
  • 22. Diblík, J., Khusainov, D.. Ya.., Baštinec, J., Sirenko, A.S.: Stability and exponential stability of linear discrete systems with constant...
  • 23. Li, H., Zhao, N., Wang, X., Zhang, X., Shi, P.: Necessary and sufficient conditions of exponential stability for delayed linear discrete-time...
  • 24. Diblík, J., Morávková, B.: Discrete matrix delayed exponential for two delays and its property. Adv. Difference Equ. 139, 1–18 (2013)
  • 25. Jin, X., Wang, J., Shen, D.: Convergence analysis for iterative learning control of impulsive linear discrete delay systems. J. Differ....
  • 26. Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005)
  • 27. Graf, U.: Applied Laplace Transforms and z-transforms for Scientists and Engineers. A Computational Approach Using a Mathematica Package,...
  • 28. Bóna, M.: A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, 2nd edn. World Scientific Pub. Co., London (2006)
  • 29. Agarwal, R. P.: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Marcel Dekker, New York, (2000)
  • 30. Medina, R., Pinto, M.: Nonlinear discrete inequalities and stability of difference equations. World Sci. Ser. Appl. Anal. 3, 467–482 (1994)
  • 31. Pachpatte, B.G.: Inequalities for Finite Difffference Equations. Dekker Inc, New York (2002)