American Institute of Mathematical Sciences

September  2020, 25(9): 3651-3657. doi: 10.3934/dcdsb.2020077

Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps

 Department of Mathematics, Ariel University, Ariel 40700, Israel

Received  October 2019 Published  January 2020

Delay differential equation is considered under stochastic perturbations of the type of white noise and Poisson's jumps. It is shown that if stochastic perturbations fade on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay provide also asymptotic mean square stability of the zero solution of the stochastic differential equation. Stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.

Citation: Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3651-3657. doi: 10.3934/dcdsb.2020077
References:
 [1] I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer, Berlin, 1972.  Google Scholar [2] E. Haynsworth, On the schur complement, Basel Mathematical Notes, 20 (1968), 17 pages. Google Scholar [3] L. Shaikhet, Lyapunov functionals and Stability of Stochastic Functional Differential Equations, Springer, Dordrecht, Heidelberg, New York, London, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar [4] 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.  Google Scholar [5] 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.  Google Scholar

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References:
 [1] I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer, Berlin, 1972.  Google Scholar [2] E. Haynsworth, On the schur complement, Basel Mathematical Notes, 20 (1968), 17 pages. Google Scholar [3] L. Shaikhet, Lyapunov functionals and Stability of Stochastic Functional Differential Equations, Springer, Dordrecht, Heidelberg, New York, London, 2013. doi: 10.1007/978-3-319-00101-2.  Google Scholar [4] 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.  Google Scholar [5] 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.  Google Scholar
trajectories (blue) of the equation equation (1.2) solution, $x(0) = 2$, $a = 0.8$, $\sigma(t) = \sqrt{3}\sin(t)$, $\gamma(t, u) = 0$, $\rho(t) = 3\sin^2(t)$ (red)
50 trajectories (blue) of the equation equation (1.2) solution, $x(0) = 2$, $a = 0.8$, $\sigma(t) = \sqrt{\dfrac{3}{t+1}}$, $\gamma(t, u) = 0$, $\rho(t) = \dfrac{3}{t+1}$ (red)
50 trajectories (blue) of the equation equation (1.2) solution, $x(0) = 2$, $a = 0.8$, $\sigma(t) = \dfrac{\sqrt{3}}{t+1}$, $\gamma(t, u) = 0$, $\rho(t) = \dfrac{3}{(t+1)^2}$ (red)
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