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  September 2020 Early access  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 and 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.

[2]

E. Haynsworth, On the schur complement, Basel Mathematical Notes, 20 (1968), 17 pages.

[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.

[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.

[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.

show all references

References:
[1]

I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Springer, Berlin, 1972.

[2]

E. Haynsworth, On the schur complement, Basel Mathematical Notes, 20 (1968), 17 pages.

[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.

[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.

[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.

Figure 1.1.  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)
Figure 1.2.  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)
Figure 1.3.  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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