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

Leonid Shaikhet

doi:10.3934/dcdsb.2021043

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2022, Volume 27, Issue 1: 301-310. Doi: 10.3934/dcdsb.2021043

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

Leonid Shaikhet

Department of Mathematics, Ariel University, Ariel 40700, Israel

Received: July 2020

Revised: November 2020

Early access: February 2021

Published: January 2022

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Abstract

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.

Keywords:

Difference equation with continuous time,
additive fading noise,
asymptotically mean square trivial,
Lyapunov functional,
Linear Matrix Inequality (LMI).

Mathematics Subject Classification: Primary: 39A30, 39A50; Secondary: 60G52.

Citation:

Full Text(HTML)

Figure 1. 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)

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Figure 2. The same as Fig. 1 besides of $ b_1 = 0.45 $

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Figure 3. The same as Fig. 1 besides of $ \sigma(t) = \dfrac{m}{\sqrt{t+\tau-m}} $, $ t>m $

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