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Exponential Stability of Stochastic Differential Delay Equations
, 1994
"... : In this paper we study both pth moment and almost sure exponential stability of the stochastic differential delay equation dx(t)=f(t;x(t);x(t\Gammaø))dt+g(t;x(t);x(t\Gammaø))dw(t). Introduce the corresponding stochastic differential equation (without delay) dx(t)=f(t;x(t);x(t))dt+ g(t;x(t);x(t))dw ..."
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Cited by 114 (46 self)
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: In this paper we study both pth moment and almost sure exponential stability of the stochastic differential delay equation dx(t)=f(t;x(t);x(t\Gammaø))dt+g(t;x(t);x(t\Gammaø))dw(t). Introduce the corresponding stochastic differential equation (without delay) dx(t)=f(t;x(t);x(t))dt+ g(t;x(t);x(t))dw(t) and assume it is exponentially stable which is guaranteed by the existence of the Lyapunov function. We shall show that the original stochastic differential delay equation remains exponentially stable provided the time lag ø is sufficiently small, and a bound for such ø is obtained. Key Words: stochastic differential delay equations, Lyapunov function, Lyapunov exponent, BorelCantelli lemma. AMS 1991 Classifications: 60H10, 34K30 1. Introduction In many branches of science and industry stochastic differential delay equations have been used to model the evolution phenomena because the measurements of timeinvolving variables and their dynamics usually contain some delays (cf. Kolmanov...
The PoincaréBendixson Theorem for Monotone Cyclic Feedback Systems with Delay
 JOURNAL OF DIFFERENTIAL EQUATIONS
, 1996
"... We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integervalued) Lyapunov function, we prove that the PoincaréBendixson theorem holds for such systems. We also obtain results on ..."
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Cited by 109 (8 self)
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We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integervalued) Lyapunov function, we prove that the PoincaréBendixson theorem holds for such systems. We also obtain results on piecewise monotonicity and stability of periodic solutions of such systems.
Robust SampledData Stabilization of Linear Systems: An Input Delay Approach
 AUTOMATICA
, 2004
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Numerical bifurcation analysis of delay differential equations
 J. Comput. Appl. Math. 125
"... We describe DDEBIFTOOL, a Matlab package for numerical bifurcation analysis of systems of delay differential equations with several fixed, discrete delays. The package implements continuation of steady state solutions and periodic solutions and their stability analysis. It also computes and continu ..."
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Cited by 82 (6 self)
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We describe DDEBIFTOOL, a Matlab package for numerical bifurcation analysis of systems of delay differential equations with several fixed, discrete delays. The package implements continuation of steady state solutions and periodic solutions and their stability analysis. It also computes and continues steady state fold and Hopf bifurcations and, from the latter, it can switch to the emanating branch of periodic solutions. We describe the numerical methods upon which the package is based and illustrate its usage and capabilities through analysing three examples: two models of coupled neurons with delayed feedback and a model of two oscillators coupled with delay. Categories and Subject Descriptors: G.1.0 [Numerical analysis]: General—numerical algorithms;
The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems
 J. Dynam. Differential Equations
, 1997
"... We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solution ..."
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Cited by 74 (5 self)
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We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c 6= 0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c 6= 0. Convergence results for solutions are obtained at the singular perturbation limit c ! 0. 1 Introduction We are interested in lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, such as the Ddimensional integer lattice Z D . Our focus in this paper is the global structure of the set of traveling wave solutions for such systems. This entails results on existence and uniqueness, and on continuous (or smooth) dependence of traveling waves and their speeds on parameters, as well as some delicate convergence results in the singular perturbation case c ! 0 of the wav...
The Fredholm Alternative for Functional Differential Equations of Mixed Type
 J. Dynam. Differential Equations
, 1999
"... We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such syst ..."
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Cited by 68 (3 self)
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We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.
A UtilityBased Congestion Control Scheme for InternetStyle Networks with Delay
, 2003
"... In this paper, we develop, analyze and implement a congestion control scheme obtained in a noncooperative game framework where each user's cost function is composed of a pricing function, proportional to the queueing delay experienced by the user, and a fairly general utility function which cap ..."
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Cited by 65 (15 self)
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In this paper, we develop, analyze and implement a congestion control scheme obtained in a noncooperative game framework where each user's cost function is composed of a pricing function, proportional to the queueing delay experienced by the user, and a fairly general utility function which captures the user demand for bandwidth. Using a network model based on fluid approximations and through a realistic modeling of queues, we establish the existence of a unique equilibrium as well as its global asymptotic stability for a general network topology. We also provide sufficient conditions for system stability when there is a bottleneck link shared by multiple users experiencing nonnegligible communication delays. Based on these theoretical foundations, we implement a windowbased, endtoend congestion control scheme, and simulate it in ns2 network simulator on various network topologies with sizable propagation delays.
A descriptor system approach to H∞ control of linear timedelay systems
 IEEE Transactions on Automatic Control
, 2002
"... Abstract—The outputfeedback control problem is solved for continuoustime, linear, retarded and neutral type systems. A delaydependent solution is obtained in terms of linear matrix inequalities (LMIs) by using a descriptor model transformation of the system and by applying Park’s inequality for ..."
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Cited by 61 (7 self)
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Abstract—The outputfeedback control problem is solved for continuoustime, linear, retarded and neutral type systems. A delaydependent solution is obtained in terms of linear matrix inequalities (LMIs) by using a descriptor model transformation of the system and by applying Park’s inequality for bounding cross terms. A statefeedback solution is derived for systems with polytopic parameter uncertainties. An outputfeedback controller is then found by solving two LMIs, one of which is associated with a descriptor timedelay “innovation filter. ” The cases of instantaneous and delayed measurements are considered. Numerical examples are given which illustrate the effectiveness of the new theory. Index Terms—Delaydependent criteria, descriptor systems,control, linear matrix inequalities (LMIs), timedelay systems. I.
Global Stability of Internet Congestion Controllers with Heterogeneous Delays
, 2004
"... In this paper, we study the problem of designing globally stable, scalable congestion control algorithms for the Internet. Prior work has primarily used linear stability as the criterion for such a design. Global stability has been studied only for single node, single source problems. Here, we obtai ..."
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Cited by 57 (1 self)
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In this paper, we study the problem of designing globally stable, scalable congestion control algorithms for the Internet. Prior work has primarily used linear stability as the criterion for such a design. Global stability has been studied only for single node, single source problems. Here, we obtain conditions for a general topology network accessed by sources with heterogeneous delays. We obtain a sufficient condition for global stability in terms of the increase/decrease parameters of the congestion control algorithm and the price functions used at the links.