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Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations

Year 2024, , 71 - 80, 14.04.2024
https://doi.org/10.36753/mathenot.1320286

Abstract

In this paper, the problem of asymptotic stability of a kind of nonlinear perturbed neutral differential system with variable delay is discussed. The Lyapunov-Krasovskii functional constructed, is used to obtain conditions for asymptotic stability of the nonlinear perturbed neutral differential system in terms of linear matrix inequality (LMI). The two new results (delay-independent and delay-dependent criteria) include and extend the existing results in the literature. Finally, an example of delay-dependent criteria is supplied and the simulation result is shown to justify the effectiveness and reliability of the used techniques.

References

  • [1] Ademola, A. T., Arawomo, P. O.: Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order. Mathematical Journal of Okayama University. 55, 157-166 (2013).
  • [2] Ademola, A. T., Arawomo, P. O., Ogunlaran, O. M., Oyekan, E. A.: Uniform stability, boundedness and asymptotic behaviour of solutions of some third order nonlinear delay differential equations. Differential Equations and Control Processes, (4), 43-66 (2013).
  • [3] Ademola, A. T., Moyo, S., Ogundiran, M. O. Arawomo, P. O., Adesina, O. A.: Stability and boundedness of solution to a certain second-order non-autonomous stochastic differential equation. International Journal of Analysis. (2016) http://dx.doi.org/10.1155/2016/2012315.
  • [4] Ademola, A. T., Ogundare, B. S., Ogundiran, M. O., Adesina, O. A.: Periodicity, stability and boundedness of solutions to certain second-order delay differential equations. International Journal of Differential equations. Article ID 2843709, 10 pages (2016).
  • [5] Ademola, A. T.: Asymptotic behaviour of solutions to certain nonlinear third order neutral functional differential equation. Heliyon 7, 1-8 (2021).
  • [6] Ademola, A. T.: Periodicity, stability, and boundedness of solutions to certain fourth order delay differential equations. International Journal of Nonlinear Science. 28(1), 20-39 (2019).
  • [7] Tejumola, H. O. Tchegnani, B.: Stability, boundedness and existence of periodic solutions of some third order and fourth-order nonlinear delay differential equations. Journal of the Nigerian Mathematical Society. 19, 9-19 (2000).
  • [8] Tunç, C.: A boundedness criterion for fourth-order nonlinear ordinary differential equations with delay. International Journal of Nonlinear Science. 6, 195-201 (2008).
  • [9] Tunç, C.: On stability of solutions of certain fourth order delay differential equations. Applied Mathematics and Mechanics (English Edition). 27, 1141-1148 (2006).
  • [10] Bellman, R., Cooke, K. L.: Differential-Difference Equations. Academic Press, New York, (1963).
  • [11] Brayton, R. K.: Nonlinear oscillations in a distributed network. Quarterly of Applied Mathematics. 24 (4), 289-301 (1967).
  • [12] Mirankefg, W. L.: The wave equation with a nonlinear interface condition. IBM Journal of Research and Development. 5, 2-24 (1961).
  • [13] Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations, Dordrecht: Kluwer Academic Publishers, (1992).
  • [14] Kolmanovskii, V. Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, Dodrecht, (1999).
  • [15] Kyrychko, Y. N., Blyuss, K. B., Gonzalez-Buelga, A., Hogan, S. J., Wagg, D. J.: Real-time dynamic substructuring in a coupled oscillator-pendulum system. Proceedings of the Royal Society London A. 462, 1271-1294 (2006).
  • [16] Liu, M., Dassios, I., Tzounas, G., Milano, F.: Model-independent derivative control delay compensation methods for power systems. Energies. 13, 342 (2020).
  • [17] Liu, M., Dassios, I., Tzounas, G., Milano, F.: Stability analysis of power systems with inclusion of realistic-modeling of WAMS delays. IEEE Transactions on Power Systems. 34, 627–636 (2019).
  • [18] Milano, F., Dassios, I.: Small-signal stability analysis for non-index 1 Hessenberg form systems of delay differential algebraic equations. IEEE Transactions on Circuits and Systems: Regular Papers. 63, 1521–1530 (2016).
  • [19] Hale, J. K., Infante, E. F., Tsen, F.-S. P.: Stability in linear delay equations. Journal of Mathematical Analysis and Applications. 105, 533-555 (1985).
  • [20] Hale, J., Verduyn Lunel, S. M.: Introduction to Functional Differential Equations New York: Springer-Verlag, (1993).
  • [21] Li, L. M.: Stability of linear neutral delay-differential systems. Bulletin of the Australian Mathematical Society. 38, 339-344 (1988).
  • [22] Slemrod, M., Infante, E. F.: Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues. Journal of Mathematical Analysis and Application. 38, 399-415 (1972).
  • [23] Brayton, R. K., Willoughby, R. A.: On the numerical integration of a symmetric system of difference-differential equations of neutral type. Journal of Mathematical Analysis and Applications. 18, 182-189 (1967).
  • [24] Khusainov, D. Ya., Yun’kova, E. V.: Investigation of the stability of linear systems of neutral type by the Lyapunov function method. Diff. Uravn, 24, 613-621 (1988).
  • [25] Park, J. H., Won. S.: Stability of neutral delay-differential systems with nonlinear perturbations. International Journal of Systems Science. 31 (8), 961-967 (2000).
  • [26] Feng, Y., Tu, D., Li. C., Huang, T.: Uniformly stability of impulsive delayed linear systems with impulsive time windows. Italian Journal of Pure and Applied Mathematics. 34, 213-220 (2015).
  • [27] Onasanya, B.O., Wen, S., Feng, Y., Zhang, W., Tang, N., Ademola, A.T.: Varying control intensity of synchronized chaotic system with time delay. Journal of Physics: Conference Series. 1828, 012143 (2021).
  • [28] Tunç, O.: Stability tests and solution estimates for non-linear differential equations. An International Journal of Optimization and Control: Theories & Applications. 13 (1), 92–103, (2023).
  • [29] Khargonekar, P. P., Petersen, I. R., Zhou, K.: Robust stabilization of uncertain linear systems: Quadratic stability and H1 control theory. IEEE Transactions on Automatic Control. 35, 356-361 (1990).
Year 2024, , 71 - 80, 14.04.2024
https://doi.org/10.36753/mathenot.1320286

Abstract

References

  • [1] Ademola, A. T., Arawomo, P. O.: Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order. Mathematical Journal of Okayama University. 55, 157-166 (2013).
  • [2] Ademola, A. T., Arawomo, P. O., Ogunlaran, O. M., Oyekan, E. A.: Uniform stability, boundedness and asymptotic behaviour of solutions of some third order nonlinear delay differential equations. Differential Equations and Control Processes, (4), 43-66 (2013).
  • [3] Ademola, A. T., Moyo, S., Ogundiran, M. O. Arawomo, P. O., Adesina, O. A.: Stability and boundedness of solution to a certain second-order non-autonomous stochastic differential equation. International Journal of Analysis. (2016) http://dx.doi.org/10.1155/2016/2012315.
  • [4] Ademola, A. T., Ogundare, B. S., Ogundiran, M. O., Adesina, O. A.: Periodicity, stability and boundedness of solutions to certain second-order delay differential equations. International Journal of Differential equations. Article ID 2843709, 10 pages (2016).
  • [5] Ademola, A. T.: Asymptotic behaviour of solutions to certain nonlinear third order neutral functional differential equation. Heliyon 7, 1-8 (2021).
  • [6] Ademola, A. T.: Periodicity, stability, and boundedness of solutions to certain fourth order delay differential equations. International Journal of Nonlinear Science. 28(1), 20-39 (2019).
  • [7] Tejumola, H. O. Tchegnani, B.: Stability, boundedness and existence of periodic solutions of some third order and fourth-order nonlinear delay differential equations. Journal of the Nigerian Mathematical Society. 19, 9-19 (2000).
  • [8] Tunç, C.: A boundedness criterion for fourth-order nonlinear ordinary differential equations with delay. International Journal of Nonlinear Science. 6, 195-201 (2008).
  • [9] Tunç, C.: On stability of solutions of certain fourth order delay differential equations. Applied Mathematics and Mechanics (English Edition). 27, 1141-1148 (2006).
  • [10] Bellman, R., Cooke, K. L.: Differential-Difference Equations. Academic Press, New York, (1963).
  • [11] Brayton, R. K.: Nonlinear oscillations in a distributed network. Quarterly of Applied Mathematics. 24 (4), 289-301 (1967).
  • [12] Mirankefg, W. L.: The wave equation with a nonlinear interface condition. IBM Journal of Research and Development. 5, 2-24 (1961).
  • [13] Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations, Dordrecht: Kluwer Academic Publishers, (1992).
  • [14] Kolmanovskii, V. Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, Dodrecht, (1999).
  • [15] Kyrychko, Y. N., Blyuss, K. B., Gonzalez-Buelga, A., Hogan, S. J., Wagg, D. J.: Real-time dynamic substructuring in a coupled oscillator-pendulum system. Proceedings of the Royal Society London A. 462, 1271-1294 (2006).
  • [16] Liu, M., Dassios, I., Tzounas, G., Milano, F.: Model-independent derivative control delay compensation methods for power systems. Energies. 13, 342 (2020).
  • [17] Liu, M., Dassios, I., Tzounas, G., Milano, F.: Stability analysis of power systems with inclusion of realistic-modeling of WAMS delays. IEEE Transactions on Power Systems. 34, 627–636 (2019).
  • [18] Milano, F., Dassios, I.: Small-signal stability analysis for non-index 1 Hessenberg form systems of delay differential algebraic equations. IEEE Transactions on Circuits and Systems: Regular Papers. 63, 1521–1530 (2016).
  • [19] Hale, J. K., Infante, E. F., Tsen, F.-S. P.: Stability in linear delay equations. Journal of Mathematical Analysis and Applications. 105, 533-555 (1985).
  • [20] Hale, J., Verduyn Lunel, S. M.: Introduction to Functional Differential Equations New York: Springer-Verlag, (1993).
  • [21] Li, L. M.: Stability of linear neutral delay-differential systems. Bulletin of the Australian Mathematical Society. 38, 339-344 (1988).
  • [22] Slemrod, M., Infante, E. F.: Asymptotic stability criteria for linear systems of difference-differential equations of neutral type and their discrete analogues. Journal of Mathematical Analysis and Application. 38, 399-415 (1972).
  • [23] Brayton, R. K., Willoughby, R. A.: On the numerical integration of a symmetric system of difference-differential equations of neutral type. Journal of Mathematical Analysis and Applications. 18, 182-189 (1967).
  • [24] Khusainov, D. Ya., Yun’kova, E. V.: Investigation of the stability of linear systems of neutral type by the Lyapunov function method. Diff. Uravn, 24, 613-621 (1988).
  • [25] Park, J. H., Won. S.: Stability of neutral delay-differential systems with nonlinear perturbations. International Journal of Systems Science. 31 (8), 961-967 (2000).
  • [26] Feng, Y., Tu, D., Li. C., Huang, T.: Uniformly stability of impulsive delayed linear systems with impulsive time windows. Italian Journal of Pure and Applied Mathematics. 34, 213-220 (2015).
  • [27] Onasanya, B.O., Wen, S., Feng, Y., Zhang, W., Tang, N., Ademola, A.T.: Varying control intensity of synchronized chaotic system with time delay. Journal of Physics: Conference Series. 1828, 012143 (2021).
  • [28] Tunç, O.: Stability tests and solution estimates for non-linear differential equations. An International Journal of Optimization and Control: Theories & Applications. 13 (1), 92–103, (2023).
  • [29] Khargonekar, P. P., Petersen, I. R., Zhou, K.: Robust stabilization of uncertain linear systems: Quadratic stability and H1 control theory. IEEE Transactions on Automatic Control. 35, 356-361 (1990).
There are 29 citations in total.

Details

Primary Language English
Subjects Dynamical Systems in Applications
Journal Section Articles
Authors

Adeleke Timothy Ademola 0000-0002-1036-1681

Adebayo Aderogba 0000-0002-4137-5445

Opeoluwa Lawrance Ogundipe 0009-0006-8182-8062

Gbenga Akınbo 0009-0009-7614-0310

Babatunde Oluwaseun Onasanya 0000-0002-3737-4044

Early Pub Date January 29, 2024
Publication Date April 14, 2024
Submission Date June 27, 2023
Acceptance Date January 24, 2024
Published in Issue Year 2024

Cite

APA Ademola, A. T., Aderogba, A., Ogundipe, O. L., Akınbo, G., et al. (2024). Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations. Mathematical Sciences and Applications E-Notes, 12(2), 71-80. https://doi.org/10.36753/mathenot.1320286
AMA Ademola AT, Aderogba A, Ogundipe OL, Akınbo G, Onasanya BO. Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations. Math. Sci. Appl. E-Notes. April 2024;12(2):71-80. doi:10.36753/mathenot.1320286
Chicago Ademola, Adeleke Timothy, Adebayo Aderogba, Opeoluwa Lawrance Ogundipe, Gbenga Akınbo, and Babatunde Oluwaseun Onasanya. “Asymptotic Stability of Neutral Differential Systems With Variable Delay and Nonlinear Perturbations”. Mathematical Sciences and Applications E-Notes 12, no. 2 (April 2024): 71-80. https://doi.org/10.36753/mathenot.1320286.
EndNote Ademola AT, Aderogba A, Ogundipe OL, Akınbo G, Onasanya BO (April 1, 2024) Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations. Mathematical Sciences and Applications E-Notes 12 2 71–80.
IEEE A. T. Ademola, A. Aderogba, O. L. Ogundipe, G. Akınbo, and B. O. Onasanya, “Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations”, Math. Sci. Appl. E-Notes, vol. 12, no. 2, pp. 71–80, 2024, doi: 10.36753/mathenot.1320286.
ISNAD Ademola, Adeleke Timothy et al. “Asymptotic Stability of Neutral Differential Systems With Variable Delay and Nonlinear Perturbations”. Mathematical Sciences and Applications E-Notes 12/2 (April 2024), 71-80. https://doi.org/10.36753/mathenot.1320286.
JAMA Ademola AT, Aderogba A, Ogundipe OL, Akınbo G, Onasanya BO. Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations. Math. Sci. Appl. E-Notes. 2024;12:71–80.
MLA Ademola, Adeleke Timothy et al. “Asymptotic Stability of Neutral Differential Systems With Variable Delay and Nonlinear Perturbations”. Mathematical Sciences and Applications E-Notes, vol. 12, no. 2, 2024, pp. 71-80, doi:10.36753/mathenot.1320286.
Vancouver Ademola AT, Aderogba A, Ogundipe OL, Akınbo G, Onasanya BO. Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations. Math. Sci. Appl. E-Notes. 2024;12(2):71-80.

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