Differential equations of second order with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. The presence of random variables in the ordinary differential equation leads to a total change of the behavior of the solution. It is not a function as in the case of deterministic equations, it is a stochastic process. It requires combining of the results in Theory of Differential Equations and Probability Theory. The initial value problem is set up in appropriate way. Sample path solutions are defined as a solutions of ordinary differential equations with determined fixed moments of impulses. P-moment generalized exponential stability is defined and some sufficient conditions for this type of stability are obtained. The study is based on the application of Lyapunov functions. The results are illustrated on examples.
Citation: Snezhana Hristova, Kremena Stefanova. p-moment exponential stability of second order differential equations with exponentially distributed moments of impulses[J]. AIMS Mathematics, 2021, 6(3): 2886-2899. doi: 10.3934/math.2021174
Differential equations of second order with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. The presence of random variables in the ordinary differential equation leads to a total change of the behavior of the solution. It is not a function as in the case of deterministic equations, it is a stochastic process. It requires combining of the results in Theory of Differential Equations and Probability Theory. The initial value problem is set up in appropriate way. Sample path solutions are defined as a solutions of ordinary differential equations with determined fixed moments of impulses. P-moment generalized exponential stability is defined and some sufficient conditions for this type of stability are obtained. The study is based on the application of Lyapunov functions. The results are illustrated on examples.
| [1] | R. Agarwal, S. Hristova, D. O'Regan, Exponential stability for differential equations with random impulses at random times, Adv. Diff. Eq., 2013 (2013), 372, Available from: https://doi.org/10.1186/1687-1847-2013-372. |
| [2] |
J. Alaba, B. Ogundare, On stability and boundedness properties of solutions of certain second order non-autonomuos nonlinear ordinary differential equations, Kragujevac J. Math., 39 (2015), 255–266. doi: 10.5937/KgJMath1502255A
|
| [3] |
M. Gowrisankar, P. Mohankumar, A. Vinodkumar, Stability results of random impulsive semilinear differential equations, Acta Math. Sci., 34 (2014), 1055–1071. doi: 10.1016/S0252-9602(14)60069-2
|
| [4] | M. Qarawani, Boundedness and asymptotic behaviour of solutions of a second order nonlinear differential equation, J. Math. Res., 4 (2012), 121–127. |
| [5] |
Z. Li, X. Shu, F. Xu, The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem, AIMS Math., 5 (2020), 6189–6210. doi: 10.3934/math.2020398
|
| [6] |
J. Sanz-Serna, A. Stuart, Ergodicity of dissipative differential equations subject to random impulses, J. Diff. Equ., 155 (1999), 262–284. doi: 10.1006/jdeq.1998.3594
|
| [7] | L. Shu, X. Shu, Q. Zhu, F. Xu, Existence and exponential stability of mild solutions for second-order neutral stochastic functional differential equation with random impulses, J. Appl. Anal. Comput., 2020. |
| [8] | Y. Tang, X. Xing, H. Karimi, L. Kocarev, J. Kurths, Tracking control of networked multiagent systems under new characterizations of impulses and its Applications in robotic systems, IEEE Trans. Ind. Electron., 63, (2016), 1299–1307. |
| [9] | A. Vinodkumar, Existence and uniqueness of solutions for random impulsive differential equation, Malaya J. Matematik, 1 (2012), 8–13. |
| [10] |
Y. Wang, X. Liu, J. Xiao, Y. Shen, Output formation-containment of interacted heterogeneous linear systems by distributed hybrid active control, Automatica, 93 (2018), 26–32. doi: 10.1016/j.automatica.2018.03.020
|
| [11] |
Y. Wang, J. Zhang, M. Liu, Brief Paper-Exponential stability of impulsive positive systems with mixed time-varying delays, IET Control Theory Appl., 8 (2014), 1537–1542. doi: 10.1049/iet-cta.2014.0231
|
| [12] |
S. Wu, X. Guo, S. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta Math. Appl. Sin., 22 (2006), 627–632. doi: 10.1007/s10255-006-0336-1
|
| [13] | S. Wu, D. Hang, X. Meng, p-Moment Stability of Stochastic Equations with Jumps, Appl. Math. Comput., 152 (2004), 505–519. |
| [14] |
H. Wu, J. Sun, p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching, Automatica, 42 (2006), 1753–1759. doi: 10.1016/j.automatica.2006.05.009
|
| [15] |
J. Yang, A. Zhong, W. Luo, Mean square stability analysis of impulsive stochastic differential equations with delays, J. Comput. Appl. Math., 216 (2008), 474–483. doi: 10.1016/j.cam.2007.05.022
|
| [16] | Y. Zhao, H. Chen, Multiplicity of solutions to two-point boundary value problems for second-order impulsive differential equations, Appl. Math. Comput., 206 (2008), 925–931. |
| [17] | S. Zhang, W. Jiang, The existence and exponential stability of random impulsive fractional differential equations, Adv. Difference Equ., 404 (2018), 17. |
| [18] | S. Zhang, J. Sun, Stability analysis of second-order differential systems with Erlang distribution random impulses, Adv. Difference Equ., 4 (2013), 10. |
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Snezhana Hristova, Kremena Stefanova. p-moment exponential stability of second order differential equations with exponentially distributed moments of impulses[J]. AIMS Mathematics, 2021, 6(3): 2886-2899. doi: 10.3934/math.2021174
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