Controllability and Hyers-Ulam stability of impulsive second order abstract damped differential systems. (English) Zbl 07905171
Summary: In this paper, we consider system of damped second order abstract impulsive differential equations to investigate its controllability and Hyers-Ulam stability. For our results about the controllability, we utilized the theory of strongly continuous cosine families of linear operators combined with Sadovskii fixed point theorem. In addition, different types of Hyers-Ulam stability is established with the help of Grönwall’s type inequality and Lipschitz conditions. At last, we give an example of damped wave equation which outline the application of our principle results.
MSC:
| 26A33 | Fractional derivatives and integrals |
| 34A08 | Fractional ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
Keywords:
damped differential equation; impulsive system; controllability; \( \beta \)-Hyers-Ulam-Rassias stabilityReferences:
| [1] | G. Arthi, and K. Balachandran, Controllability of damped second-order neutral functional differential systems with impulses, Indian J. Pure Appl. Math., 2012, 16(1), 89-106. · Zbl 1235.93042 |
| [2] | T. Hakon Gronwall, Note on the derivatives with respect to a parameter of the solution of a system of differential equations, Ann. of Math., 1919, 20(2), 292-296. · JFM 47.0399.02 |
| [3] | D. H. Hyers, G. Isac and T. M. Rassias, Stability of functional equations in several variables, Adv. Appl. Clifford Algebr., 1998. · Zbl 0907.39025 |
| [4] | D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 1941, 27, 222-224. · Zbl 0061.26403 |
| [5] | E. Hernandez, K. Balachandran and N. Annapoorani, Existance results for a damped second order abstract functional differential equation with impulses. Math. Comput. Modelling, 2009, 50, 1583-1594. · Zbl 1185.34113 |
| [6] | T. Li and A. Zada, Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ, 2016, 153. · Zbl 1419.39038 |
| [7] | T. Li, A. Zada, and S. Faisal, Hyers-Ulam stability of nth order linear differ-ential equations, J. Nonlinear Sci. Appl, 2016, 9, 2070-2075. · Zbl 1337.34052 |
| [8] | Y. Lin and N. Tanaka, Nonlinear abstract wave equations with strong damping,z J. Math. Anal. Appl, 1998, 225, 46-61. · Zbl 0918.35094 |
| [9] | V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. · Zbl 0719.34002 |
| [10] | Y. Liu and D. O’Regan, Controllability of impulsive functional differential sys-tems with nonlocal conditions, Electron. J. Diff. Equ. 2013, 194, 1-10. · Zbl 1288.34073 |
| [11] | M. Muslim, A. Kumar and M. Feckan, Existence, uniqueness and stability of solutions to second order nonlinear differential equation with non-instantaneous impulses, Journal of King Saud, 2018, 30, 204-213. |
| [12] | S. K. Ntouyas and D. O’Regan, Some remarks on controllability of evolution equation in Banach space, Electron. J. Differ. Equ. Monogr, 2009, 79, 1-6. · Zbl 1168.93005 |
| [13] | J. Nieto, and D. O’Regan, Variational approach to impulsive differential equa-tions, Nonlinear Anal. Real World Appl, 2009, 10(2), 680-690. · Zbl 1167.34318 |
| [14] | J. Y. Park and H. K. Han, Controllability for some second order differential equations, Bull. Korean Math. Soc, 1997, 34, 411-419. · Zbl 0889.93008 |
| [15] | H. Qin, Z. Gu, Y. Fu, and T. Li, Existence of mild solutions and controllability of fractional impulsive integro differential systems with nonlocal conditions, J. Funct. Spaces, 2017, 1-11. · Zbl 1377.45008 |
| [16] | Y. V. Rogovchenko, Impulsive evolution systems: Main results and new trends, Dyn. Contin. Discrete Impuls Syst, 1997, 3, 57-88. · Zbl 0879.34014 |
| [17] | T. M. Rassias, On the stability of the linear mapping in Banach spaces, P. Am. Math. Soc, 1978, 72(2), 297-300. · Zbl 0398.47040 |
| [18] | A. M. Samoilenko and N. A. Perestyuk, Stability of solutions of differential equations with impulse effect, J. Differ. Equations, 1977, 13, 1981-1992. |
| [19] | B. N. Sadovskii, On a fixed point principle, Funct. Anal. Appl, 1967, 1, 74-76. · Zbl 0165.49102 |
| [20] | A. M. Samoilenka and N. A. Perestyunk, Impulsive Differential Equations, World Scientific, Singapore, 1995. · Zbl 0837.34003 |
| [21] | M. A. Shubov, C. F. Martin, J. P. Dauer and B. Belinskii, Exact controllability of damped wave equation, SIAM J. Control Optim, 1997, 35, 1773-1789. · Zbl 0889.35057 |
| [22] | S. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh and T. Li, Stability of higher-order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl, 2016, 9, 4713-4721. · Zbl 1350.34022 |
| [23] | S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, Interscience Publishers, New York, 1960. · Zbl 0086.24101 |
| [24] | P. Wang, C. Li, J. Zhang, and T. Li, Quasilinearization method for first-order impulsive integro-differential equations, Electron. J. Differential Equa-tions, 2019, 1-14. · Zbl 1442.45009 |
| [25] | J. Wang, M. Fečkan and Y. Zhou, Ulam’s type stability of impulsive deferential equations, J. Math. Ana. Appl, 2012, 395(1), 258-264. · Zbl 1254.34022 |
| [26] | J. Wang, A. Zada and W. Ali, Ulam’s-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlin. Sci. Num, 2018, 19(5), 553-560. · Zbl 1401.34091 |
| [27] | J. Wang, M. Fečkan and Y. Tian, Stability analysis for a general class of non-instantaneous impulsive differential equations, Mediter. J. Math, 2017, 14, 1-21. · Zbl 1373.34031 |
| [28] | D. Xu, and Y. Zhichun, Impulsive delay differential inequality and stability of neural networks. J. Math. Anal. Appl., 2005, 305(1), 107-120. · Zbl 1091.34046 |
| [29] | X. Xu, Y. Liu, H. Li and F. E. Alsaadi, Synchronization of switched boolean networks with impulsive effects, Int. J. Biomath, 2018, 11, 1850080. · Zbl 1397.93106 |
| [30] | X. Yu, J. Wang and Y. Zhang, On the β-Ulam-Rassias stability of nonaut-nonomous impulsive evolution equations, J. Appl. Math. Comput, 2015, 48, 461-475. · Zbl 1321.34075 |
| [31] | A. Zada, W. Ali and C. Park, Ulam’s type stability of higher order nonlinear de-lay differential equations via integral inequality of Grönwall-Bellman-Bihari’s type, Appl. Math. Comput, 2019, 350, 60-65. · Zbl 1428.34087 |
| [32] | A. Zada, P. Wang, D. Lassoued, and T. Li, Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems, Adv. Difference Equ, 2017, 192. · Zbl 1422.34172 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
