Mean-square stability analysis for nonlinear stochastic pantograph equations by transformation approach

Yang, Huizi; Yang, Zhanwen; Wang, Pengyue; Han, Dan

doi:10.1016/j.jmaa.2019.06.061

Cited by 13 publications

(13 citation statements)

References 12 publications

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“…If Φ(t, u, v) � K(u + v), the non-Lipschitz conditions reduce to a global Lipschitz conditions which have been investigated in previous literatures [5][6][7][8]. So, some previous results [5][6][7][8] will be significantly improved in our paper. Also, the averaging principle is applied to study stochastic pantograph equations for the first time.…”

Section: Resultsmentioning

confidence: 84%

“…Based on these irreplaceable roles, the existence, uniqueness, and stability for different kinds of pantograph equations were tremendously investigated by many scholars. Of course, some excellent and important articles have also emerged in our vision (see [5][6][7][8] and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…(i) We first attempt to investigate the property of solutions for a class of stochastic pantograph equations by the averaging principle under the non-Lipschitz conditions. Comparing with some previous literatures [5][6][7][8], the corresponding conditions are required to satisfy the Lipschitz condition or the local Lipschitz condition. However, in some practical cases, the Lipschitz condition is usually violated.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

Gao

Luo

et al. 2021

Mathematical Problems in Engineering

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This paper is devoted to presenting an averaging principle for stochastic pantograph equations. Under suitable non-Lipschitz conditions, the solutions to stochastic pantograph equations can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability. At last, an example is given to demonstrate the feasibility of obtained results. Moreover, our results have generalized significantly some previous ones.

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Section: Resultsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

Gao

Luo

et al. 2021

Mathematical Problems in Engineering

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“…Definition 1 (31). System (1) is said to be asymptotically mean square stable if, for any (x) and r 0 ∈ , the system state y(x, t) satisfies lim t→∞ E||y(⋅, t)|| 2 = 0.…”

Section: Mathematical Model and Preliminariesmentioning

confidence: 99%

Boundary control of stochastic reaction‐diffusion systems with Markovian switching

Han

Ding

et al. 2020

Intl J Robust & Nonlinear

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This article focuses on the boundary control of stochastic Markovian reaction-diffusion systems (SMRDSs). Both the cases of completely known and partially unknown transition probabilities are taken into account. By using the Lyapunov functional method, a sufficient condition is obtained under the designed boundary controllers to guarantee the asymptotic mean square stability for SMRDSs with completely known transition probabilities. For the case of partially unknown transition probabilities, we introduce free-connection weighting matrices to handle the boundary control problem. When external disturbance enters the system, a sufficient criterion of H-infinity boundary control is developed. Furthermore, robust stabilization is investigated for parametric uncertain SMRDSs in both cases. Two examples are presented to demonstrate the efficiency of the proposed approaches. K E Y W O R D S asymptotic stability, boundary control, H-infinity control, partially unknown transition probabilities, stochastic Markovian reaction-diffusion systems Int J Robust Nonlinear Control. 2020;30:4129-4148.wileyonlinelibrary.com/journal/rnc

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“…An important special case is the linear delay function θ(t) = qt (0 < q < 1), which is also known as the pantograph case (Refs. Brunner et al 2010;Fox et al 1971;Iserles 1993Iserles , 1994Iserles , 1997Yang et al 2019). It is natural to extend the discussion to a more general form of nonlinear functional equations and discuss their numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

Analysis of continuous collocation solutions for nonlinear functional equations with vanishing delays

et al. 2019

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In this paper, we study the existence, uniqueness, and regularity of solutions for the nonlinear functional equations y(t) = b(t, y(θ (t))) + f (t), t ∈ [0, T ], with vanishing delay θ(t). We then present a collocation method to solve this equation, and analyze the convergence properties of the piecewise polynomial collocation approximations. Several numerical experiments are used to illustrate the theoretical results.

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Mean-square stability analysis for nonlinear stochastic pantograph equations by transformation approach

Cited by 13 publications

References 12 publications

The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

The Averaging Principle for Stochastic Pantograph Equations with Non-Lipschitz Conditions

Boundary control of stochastic reaction‐diffusion systems with Markovian switching

Analysis of continuous collocation solutions for nonlinear functional equations with vanishing delays

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