Mathematical Theories and Applications for Nonlinear Control SystemsView this Special Issue
Research Article | Open Access
Jing Zhao, Fanwei Meng, "Stability Analysis of Solutions for a Kind of Integro-Differential Equations with a Delay", Mathematical Problems in Engineering, vol. 2018, Article ID 9519020, 6 pages, 2018. https://doi.org/10.1155/2018/9519020
Stability Analysis of Solutions for a Kind of Integro-Differential Equations with a Delay
Stability of zero solution for second-order integro-differential equations with a delay is analyzed and some new results are presented. Through constructing Lyapunov functional, we give the corresponding sufficient conditions on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to support our new results.
Stability and the existence of solutions for nonlinear differential equations have been studied by many scholars [1–20] due to their many applications to problems in information theory, control theory, mechanics, chemistry, physics, and so on. In , the authors considered a second-order functional integro-differential equation with multiple delaysand they gave some new conditions on the continuability and boundedness of solutions. Li  studied -stability of the trivial solutions of the following three nonlinear systems with time delay,and two Volterra integro-differential equationsIn , Afuwape studied the asymptotic stability and the uniformly ultimate boundedness of the solutions for a kind of third-order delay differential equations and gave some sufficient conditions.
This paper investigates second-order integro-differential equations with a delay and obtains some new results on the stability of zero solution. By constructing Lyapunov functional, the corresponding sufficient conditions are present on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to show that our results are effective.
2. Main Results
Consider the following integro-differential equation with a delay:where , is a constant, , and are continuous with , and with and .
We can rewrite (5) as the system
Theorem 1. Consider system (6). There exist nonnegative constants , , , , and , a positive number , and functions , such that the following conditions hold:() when , .() , when .() , , .() . Then the zero solution of system (6) is stable.
Proof. Define a Lyapunov functional aswhere is a positive constant to be determined. Using condition (), we haveand .
Suppose that is a solution of (6). ThenBy conditions () and () and , we haveThen,Choosing , we haveTherefore, the zero solution of system (6) is stable and the proof is completed.
Theorem 1 can be generalized to the form with a variable delay . It only needs a new condition. Then, the following result is obtained.
Corollary 2. Consider system (6) with a variable delay . Conditions (1)-(3) of Theorem 1 are satisfied. Moreover,() ,() there are and , such that and . Then the zero solution of system (6) with a variable delay is stable.
Next, we consider another second-order integro-differential system with time delay
System (16) can be rewritten as follows:
Theorem 3. Consider system (17). There are nonnegative constants , , , , and , positive numbers , , and , and functions and , such that the conditions hold:() when , .() , , , when .() , , , .() .() . Then the zero solution of system (17) is stable.
Proof. Define a Lyapunov functional aswhere and are two positive constants to be determined. Using condition (), we haveand then .
Let be a solution of (17). Then,Applying conditions () and () and , we haveBy condition (), we haveThus, we choose and . Using conditions () and (), we haveTherefore, the zero solution of system (17) is stable and the proof is completed.
Corollary 4. Consider system (17) with a variable delay . Conditions (1)-(3) of Theorem 3 are satisfied. Moreover,() ,() there are and , such that and . Then the zero solution of system (17) with a variable delay is stable.
3. An Illustrative Example
In this section, we give an illustrative example to the effectiveness of results obtained in this paper. Moreover, we list a graph of solutions of an integral-differential equation with a delay to verify the correctness of the conclusion.
Example 1. Consider the following example:
In the example, , , , , , and a delay . Obvious, this system is the same form as (5). Moreover,() , ;() , ;() , , ;() = , .
Thus, all conditions of Theorem 1 are satisfied and the zero solution of system (27) is stable by the obtained result. To show the effectiveness of the result, we carry out a simulation result with the following choices. Initial Condition: for . The simulation result is shown in Figure 1, which is the stability of solutions for system (27).
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The first author is the main completer. The second author is the corresponding author and provides the thought of this paper and advice in the process of writing.
This work is supported by the National Natural Science Foundation of China (G11671227, G11701310, and G61403223).
- A. U. Afuwape and M. O. Omeike, “On the stability and boundedness of solutions of a kind of third order delay differential equations,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 444–451, 2008.
- A. Diamandescu, “On the ψ-stability of a nonlinear Volterra integro-differential system,” Electronic Journal of Differential Equations, vol. 56, pp. 1–14, 2005.
- A. Diamandescu, “On the ψ-asymptotic stability of a nonlinear Volterra integro-differential system,” Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, vol. 46, no. 1-2, pp. 39–60, 2003.
- K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic, Dordrecht, The Netherlands, 1992.
- J. R. Graef and C. Tunç, “Continuability and boundedness of multi-delay functional integro-differential equations of the second order,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A: Matematicas, vol. 109, no. 1, pp. 169–173, 2015.
- L. Li, M. Han, X. Xue, and Y. Liu, “Ψ-stability of nonlinear Volterra integro-differential systems with time delay,” Abstract and Applied Analysis, pp. 1–5, 2013.
- H. Liu and F. Meng, “Some new nonlinear integral inequalities with weakly singular kernel and their applications to FDEs,” Journal of Inequalities and Applications, vol. 2015, no. 209, 2015.
- F. Meng, L. Li, and Y. Bai, “ψ-stability of nonlinear Volterra integro-differential systems,” Dynamic Systems and Applications, vol. 20, no. 4, pp. 563–574, 2011.
- W. Sun, L. Peng, Y. Zhang, and H. Jia, “H∞ excitation control design for stochastic power systems with input delay based on nonlinear Hamiltonian system theory,” Mathematical Problems in Engineering, vol. 2015, Article ID 947815, 12 pages, 2015.
- Weiwei Sun, Kaili Wang, Congcong Nie, and Xuejun Xie, “Energy-Based Controller Design of Stochastic Magnetic Levitation System,” Mathematical Problems in Engineering, vol. 2017, pp. 1–6, 2017.
- W. Sun and L. Peng, “Robust adaptive control of uncertain stochastic Hamiltonian systems with time varying delay,” Asian Journal of Control, vol. 18, no. 2, pp. 642–651, 2016.
- W. Sun and L. Peng, “Observer-based robust adaptive control for uncertain stochastic Hamiltonian systems with state and input delays,” Lithuanian Association of Nonlinear Analysts. Nonlinear Analysis: Modelling and Control, vol. 19, no. 4, pp. 626–645, 2014.
- X. Xie and M. Jiang, “Output feedback stabilization of stochastic feedforward nonlinear time-delay systems with unknown output function,” International Journal of Robust and Nonlinear Control, vol. 28, no. 1, pp. 266–280, 2018.
- X.-J. Xie, Z.-J. Li, and K. Zhang, “Semi-global output feedback control for nonlinear systems with uncertain time-delay and output function,” International Journal of Robust and Nonlinear Control, vol. 27, no. 15, pp. 2549–2566, 2017.
- R. Xu and Y. Zhang, “Generalized Gronwall fractional summation inequalities and their applications,” Journal of Inequalities and Applications, vol. 2015, no. 42, 2015.
- R. Xu and F. Meng, “Some new weakly singular integral inequalities and their applications to fractional differential equations,” Journal of Inequalities and Applications, vol. 2016, no. 78, 2016.
- Y. Wang and L. Liu, “Positive properties of the Green function for two-term fractional differential equations and its application,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 4, pp. 2094–2102, 2017.
- Z. Zheng, X. Gao, and J. Shao, “Some new generalized retarded inequalities for discontinuous functions and their applications,” Journal of Inequalities and Applications, vol. 7, 2016.
- Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011.
- Y. Zhao, Y. Wang, X. Zhang, and H. Li, “Feedback stabilisation control design for fractional order non-linear systems in the lower triangular form,” IET Control Theory & Applications, vol. 10, no. 9, pp. 1061–1068, 2016.
Copyright © 2018 Jing Zhao and Fanwei Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.