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

Year 2024, Volume: 12 Issue: 2, 71 - 80, 14.04.2024

Adeleke Timothy Ademola , Adebayo Aderogba , Opeoluwa Lawrance Ogundipe , Gbenga Akınbo , Babatunde Oluwaseun Onasanya

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.

Keywords

Asymptotic stability, Lyapunov-Krasovskii functional, Neutral delay differential system, Perturbation analysis

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).