Stability of delay evolution equations with stochastic perturbations
type: Item
instance of: scholarly article
title: Stability of delay evolution equations with stochastic perturbations
zbMATH Open document ID: 1316.60104
author: Leonid Shaikhet
author: T. Caraballo
published in: Communications on Pure and Applied Analysis
publication date: 2014-12-05T00:00:00Z
review text: In order to study the problem of exponential stability in the second moment of the strong solution of the time-varying delay evolution equation \[ du(t)=(A(t, u(t)) +F(u(t-h(t))))dt + B(t, u(t-\tau(t)))dW(t) \] with initial data \(u(s)=\psi(s)\) for \(s\in [-h, 0]\) in the framework of a Gelfand triple \(U\subset H=H^*\subset U^*\), the authors aim to explain two methods for the construction of a Lyapunov functional. They first remark a theorem about mean-square stability of the trivial solution of the general functional evolution equation \[ du(t)=(A(t, u(t)) + f(t, u_t))dt + B(t, u_t)dW(t) \] with initial data \(u(s)=\psi(s)\) for \(s\in [-h, 0]\) in which \(A(t,\cdot): U\to U^*\) is a family of nonlinear monotone and coercive operators. Here, of course, the concept of solution is weaker than the one introduced in similar previous papers. Then, after a long calculation, the authors obtain a sufficient condition in terms of constants and coefficients appearing in the above mentioned delay equation for the stability of the trivial solution. Finally, they apply the results of the paper to stochastic two-dimensional Navier-Stokes equations with delay. The paper concludes with a comparison between the two methods by analyzing the stability region for the parameters involved in three different reaction-diffusion equations.
reviewed by: Ruhollah Jahanipur
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35K35
zbMATH DE Number: 6377309
zbMATH Keywords: Lyapunov functional
zbMATH Keywords: stochastic delay evolution equation
zbMATH Keywords: exponential stability
zbMATH Keywords: second moment
zbMATH Keywords: stochastic Navier-Stokes equations
MaRDI profile type: Publication
full work available at URL: cpaa.2014.13.2095
OpenAlex ID: W2315610790
cites work: Asymptotic Stability of Nonlinear Stochastic Evolution Equations
cites work: Exponential stability of mild solutions of stochastic partial differential equations with delays
cites work: Stochastic functional partial differential equations: Existence, uniqueness and asymptotic decay property
cites work: Method of Lyapunov functionals construction in stability of delay evolution equations
cites work: Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays
cites work: Construction of Lyapunov functionals for stochastic hereditary systems: A survey of some recent results
cites work: Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays
cites work: Q4137839
cites work: Stochastic Equations in Infinite Dimensions
cites work: Lyapunov Functionals and Stability of Stochastic Difference Equations
cites work: Q4396394
cites work: Exponential stability of nonautonomous stochastic partial differential equations with finite memory
cites work: Q3054464
DOI: 10.3934/CPAA.2014.13.2095
description: scientific article; zbMATH DE number 6377309