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January  2022, 27(1): 301-310. doi: 10.3934/dcdsb.2021043

## Behavior of solution of stochastic difference equation with continuous time under additive fading noise

 Department of Mathematics, Ariel University, Ariel 40700, Israel

Received  July 2020 Revised  November 2020 Published  January 2022 Early access  February 2021

Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically mean square quasistable and the level of additive stochastic perturbations is given by square summable sequence, then the solution of a perturbed difference equation remains to be an asymptotically mean square quasitrivial. The obtained results are formulated in terms of Lyapunov functionals and linear matrix inequalities (LMIs). It is noted that the study of the situation, when an additive stochastic noise fades on the infinity not so quickly, remains an open problem.

Citation: Leonid Shaikhet. Behavior of solution of stochastic difference equation with continuous time under additive fading noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 301-310. doi: 10.3934/dcdsb.2021043
##### References:
 [1] S. Damak, M. Di Loreto, W. Lombardi and V. Andrieu, Exponential L2-stability for a class of linear systems governed by continuous-time difference equations, Automatica, 50 (2014), 3299-3303.  doi: 10.1016/j.automatica.2014.10.087. [2] S. Damak, M. Di Loreto and S. Mondié, Stability of linear continuous-time difference equations with distributed delay: Constructive exponential estimates, International Journal of Robust and Nonlinear Control, 25 (2015), 3195-3209.  doi: 10.1002/rnc.3249. [3] M. Di Loreto, S. Damak and S. Mondié, Stability and stabilization for continuous-time difference equations with distributed delay, Delays and Networked Control Systems, 6 (2016), 17-36. [4] M. Gil' and S. Cheng, Solution estimates for semilinear difference-delay equations with continuous time, Discrete Dynamics in Nature and Society, 2007 (2007), Article ID 82027, 8 pages. doi: 10.1155/2007/82027. [5] J. Luo and L. Shaikhet, Stability in probability of nonlinear stochastic Volterra difference equations with continuous variable, Stochastic Analysis and Applications, 25 (2007), 1151-1165.  doi: 10.1080/07362990701567256. [6] Q. Ma, K. Gu and N. Choubedar, Strong stability of a class of difference equations of continuous time and structured singular value problem, Automatica, 87 (2018), 32-39.  doi: 10.1016/j.automatica.2017.09.012. [7] D. Melchor-Aguilar, Exponential stability of some linear continuous time difference systems, Systems & Control Letters, 61 (2012), 62-68.  doi: 10.1016/j.sysconle.2011.09.013. [8] D. Melchor-Aguilar, Exponential stability of linear continuous time difference systems with multiple delays, Systems & Control Letters, 62 (2013), 811-818.  doi: 10.1016/j.sysconle.2013.06.003. [9] D. Melchor-Aguilar, Further results on exponential stability of linear continuous time difference systems, Applied Mathematics and Computation, 219 (2013), 10025-10032.  doi: 10.1016/j.amc.2013.03.051. [10] P. Pepe, The Liapunov's second method for continuous time difference equations, International Journal of Robust and Nonlinear Control, 13 (2003), 1389-1405.  doi: 10.1002/rnc.861. [11] E. Rocha, S. Mondié and M. Di Loreto, On the Lyapunov matrix of linear delay difference equations in continuous time, IFAC-PapersOnLine, 50 (2017), 6507-6512. [12] E. Rocha, S. Mondié and M. Di Loreto, Necessary stability conditions for linear difference equations in continuous time, IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 63 (2018), 4405-4412.  doi: 10.1109/TAC.2018.2822667. [13] L. Shaikhet, Lyapunov functionals construction for stochastic difference second kind Volterra equations with continuous time, Advances in Difference Equations, 2004 (2004), 67-91.  doi: 10.1155/S1687183904308022. [14] L. Shaikhet, About Lyapunov functionals construction for difference equations with continuous time, Applied Mathematics Letters, 17 (2004), 985-991.  doi: 10.1016/j.aml.2003.06.011. [15] L. Shaikhet, Construction of Lyapunov functionals for stochastic difference equations with continuous time, Mathematics and Computers in Simulation, 66 (2004), 509-521.  doi: 10.1016/j.matcom.2004.03.006. [16] L. Shaikhet, About an unsolved stability problem for a stochastic difference equation with continuous time, Journal of Difference Equations and Applications, 17 (2011), 441-444.  doi: 10.1080/10236190903489973. [17] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer Science & Busines Media, 2011. doi: 10.1007/978-0-85729-685-6. [18] Y. Zhang, Robust exponential stability of uncertain impulsive delay difference equations with continuous time, Journal of the Franklin Institute, 348 (2011), 1965-1982.  doi: 10.1016/j.jfranklin.2011.05.014. [19] L. Shaikhet, About stability of delay differential equations with square integrable level of stochastic perturbations, Applied Mathematics Letters, 90 (2019), 30-35.  doi: 10.1016/j.aml.2018.10.004. [20] L. Shaikhet, About stability of difference equations with continuous time and fading stochastic perturbations, Applied Mathematics Letters, 98 (2019), 284-291.  doi: 10.1016/j.aml.2019.06.029. [21] L. Shaikhet, About stability of difference equations with square summable level of stochastic perturbations, Journal of Difference Equations and Applications, 26 (2020), 362-369.  doi: 10.1080/10236198.2020.1734585. [22] L. Shaikhet, Stability of delay differential equations with fading stochastic perturbations of the type of white noise and Poisson's jumps, Discrete and Continuous Dynamical Systems Series B, 25 (2020), 3651-3657.  doi: 10.3934/dcdsb.2020077.

show all references

##### References:
 [1] S. Damak, M. Di Loreto, W. Lombardi and V. Andrieu, Exponential L2-stability for a class of linear systems governed by continuous-time difference equations, Automatica, 50 (2014), 3299-3303.  doi: 10.1016/j.automatica.2014.10.087. [2] S. Damak, M. Di Loreto and S. Mondié, Stability of linear continuous-time difference equations with distributed delay: Constructive exponential estimates, International Journal of Robust and Nonlinear Control, 25 (2015), 3195-3209.  doi: 10.1002/rnc.3249. [3] M. Di Loreto, S. Damak and S. Mondié, Stability and stabilization for continuous-time difference equations with distributed delay, Delays and Networked Control Systems, 6 (2016), 17-36. [4] M. Gil' and S. Cheng, Solution estimates for semilinear difference-delay equations with continuous time, Discrete Dynamics in Nature and Society, 2007 (2007), Article ID 82027, 8 pages. doi: 10.1155/2007/82027. [5] J. Luo and L. Shaikhet, Stability in probability of nonlinear stochastic Volterra difference equations with continuous variable, Stochastic Analysis and Applications, 25 (2007), 1151-1165.  doi: 10.1080/07362990701567256. [6] Q. Ma, K. Gu and N. Choubedar, Strong stability of a class of difference equations of continuous time and structured singular value problem, Automatica, 87 (2018), 32-39.  doi: 10.1016/j.automatica.2017.09.012. [7] D. Melchor-Aguilar, Exponential stability of some linear continuous time difference systems, Systems & Control Letters, 61 (2012), 62-68.  doi: 10.1016/j.sysconle.2011.09.013. [8] D. Melchor-Aguilar, Exponential stability of linear continuous time difference systems with multiple delays, Systems & Control Letters, 62 (2013), 811-818.  doi: 10.1016/j.sysconle.2013.06.003. [9] D. Melchor-Aguilar, Further results on exponential stability of linear continuous time difference systems, Applied Mathematics and Computation, 219 (2013), 10025-10032.  doi: 10.1016/j.amc.2013.03.051. [10] P. Pepe, The Liapunov's second method for continuous time difference equations, International Journal of Robust and Nonlinear Control, 13 (2003), 1389-1405.  doi: 10.1002/rnc.861. [11] E. Rocha, S. Mondié and M. Di Loreto, On the Lyapunov matrix of linear delay difference equations in continuous time, IFAC-PapersOnLine, 50 (2017), 6507-6512. [12] E. Rocha, S. Mondié and M. Di Loreto, Necessary stability conditions for linear difference equations in continuous time, IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 63 (2018), 4405-4412.  doi: 10.1109/TAC.2018.2822667. [13] L. Shaikhet, Lyapunov functionals construction for stochastic difference second kind Volterra equations with continuous time, Advances in Difference Equations, 2004 (2004), 67-91.  doi: 10.1155/S1687183904308022. [14] L. Shaikhet, About Lyapunov functionals construction for difference equations with continuous time, Applied Mathematics Letters, 17 (2004), 985-991.  doi: 10.1016/j.aml.2003.06.011. [15] L. Shaikhet, Construction of Lyapunov functionals for stochastic difference equations with continuous time, Mathematics and Computers in Simulation, 66 (2004), 509-521.  doi: 10.1016/j.matcom.2004.03.006. [16] L. Shaikhet, About an unsolved stability problem for a stochastic difference equation with continuous time, Journal of Difference Equations and Applications, 17 (2011), 441-444.  doi: 10.1080/10236190903489973. [17] L. Shaikhet, Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer Science & Busines Media, 2011. doi: 10.1007/978-0-85729-685-6. [18] Y. Zhang, Robust exponential stability of uncertain impulsive delay difference equations with continuous time, Journal of the Franklin Institute, 348 (2011), 1965-1982.  doi: 10.1016/j.jfranklin.2011.05.014. [19] L. Shaikhet, About stability of delay differential equations with square integrable level of stochastic perturbations, Applied Mathematics Letters, 90 (2019), 30-35.  doi: 10.1016/j.aml.2018.10.004. [20] L. Shaikhet, About stability of difference equations with continuous time and fading stochastic perturbations, Applied Mathematics Letters, 98 (2019), 284-291.  doi: 10.1016/j.aml.2019.06.029. [21] L. Shaikhet, About stability of difference equations with square summable level of stochastic perturbations, Journal of Difference Equations and Applications, 26 (2020), 362-369.  doi: 10.1080/10236198.2020.1734585. [22] L. Shaikhet, Stability of delay differential equations with fading stochastic perturbations of the type of white noise and Poisson's jumps, Discrete and Continuous Dynamical Systems Series B, 25 (2020), 3651-3657.  doi: 10.3934/dcdsb.2020077.
50 trajectories (green) of the solution of the equation (1.1) by $k = 1$, $\tau = 1$, $a_0 = a_1 = b_0 = 0.45$, $b_1 = -0.45$ and $\sigma(t)$ (red)
The same as Fig. 1 besides of $b_1 = 0.45$
The same as Fig. 1 besides of $\sigma(t) = \dfrac{m}{\sqrt{t+\tau-m}}$, $t>m$
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