4,126 research outputs found
General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback
In this paper we consider a viscoelastic wave equation with a time-varying
delay term, the coefficient of which is not necessarily positive. By
introducing suitable energy and Lyapunov functionals, under suitable
assumptions, we establish a general energy decay result from which the
exponential and polynomial types of decay are only special cases.Comment: 11 page
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Robust filtering for stochastic genetic regulatory networks with time-varying delay
This is the post print version of the article. The official published version can be obtained from the link - Copyright 2009 Elsevier LtdThis paper addresses the robust filtering problem for a class of linear genetic regulatory networks (GRNs) with stochastic disturbances, parameter uncertainties and time delays. The parameter uncertainties are assumed to reside in a polytopic region, the stochastic disturbance is state-dependent described by a scalar Brownian motion, and the time-varying delays enter into both the translation process and the feedback regulation process. We aim to estimate the true concentrations of mRNA and protein by designing a linear filter such that, for all admissible time delays, stochastic disturbances as well as polytopic uncertainties, the augmented state estimation dynamics is exponentially mean square stable with an expected decay rate. A delay-dependent linear matrix inequality (LMI) approach is first developed to derive sufficient conditions that guarantee the exponential stability of the augmented dynamics, and then the filter gains are parameterized in terms of the solution to a set of LMIs. Note that LMIs can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures.This work was supported in part by the Biotechnology and Biological Sciences Research Council (BBSRC) of the U.K. under Grants BB/C506264/1 and 100/EGM17735, an International Joint Project sponsored by the Royal Society of the U.K., the Research Grants Council of Hong Kong under Grant HKU 7031/06P, the National Natural Science Foundation of China under Grant 60804028, and the Alexander von Humboldt Foundation of Germany
On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay
An initial-boundary value problem for a viscoelastic wave equation subject to
a strong time-localized delay in a Kelvin & Voigt-type material law is
considered. Transforming the equation to an abstract Cauchy problem on the
extended phase space, a global well-posedness theory is established using the
operator semigroup theory both in Sobolev-valued - and BV-spaces. Under
appropriate assumptions on the coefficients, a global exponential decay rate is
obtained and the stability region in the parameter space is further explored
using the Lyapunov's indirect method. The singular limit is
further studied with the aid of the energy method. Finally, a numerical example
from a real-world application in biomechanics is presented.Comment: 34 pages, 4 figures, 1 set of Matlab code
Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation
This paper studies the asymptotic growth and decay properties of solutions of
the stochastic pantograph equation with multiplicative noise. We give
sufficient conditions on the parameters for solutions to grow at a polynomial
rate in -th mean and in the almost sure sense. Under stronger conditions the
solutions decay to zero with a polynomial rate in -th mean and in the almost
sure sense. When polynomial bounds cannot be achieved, we show for a different
set of parameters that exponential growth bounds of solutions in -th mean
and an almost sure sense can be obtained. Analogous results are established for
pantograph equations with several delays, and for general finite dimensional
equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of
Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ.
(July 1--4, 2015, Szeged, Hungary
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Filtering for nonlinear genetic regulatory networks with stochastic disturbances
In this paper, the filtering problem is investigated for nonlinear genetic regulatory networks with stochastic disturbances and time delays, where the nonlinear function describing the feedback regulation is assumed to satisfy the sector condition, the stochastic perturbation is in the form of a scalar Brownian motion, and the time delays exist in both the translation process and the feedback regulation process. The purpose of the addressed filtering problem is to estimate the true concentrations of the mRNA and protein. Specifically, we are interested in designing a linear filter such that, in the presence of time delays, stochastic disturbances as well as sector nonlinearities, the filtering dynamics of state estimation for the stochastic genetic regulatory network is exponentially mean square stable with a prescribed decay rate lower bound beta. By using the linear matrix inequality (LMI) technique, sufficient conditions are first derived for ensuring the desired filtering performance for the gene regulatory model, and the filter gain is then characterized in terms of the solution to an LMI, which can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures
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