Controlling linear functional differential equation

Necessary conditions for the optimal control of a linear system of neutral type functional differential equations are obtained. We show that a success in reducing the initial problem to the problem of solvability of a certain boundary value problem is, in principle, a question of one's ability to construct adjoint operators to the operators appearing in the initial control problem.     

Non-autonomous scalar discontinuous ordinary differential equation

In this paper we prove that the problem x^′(t)=f(x(t))+h(t), x(0)=0, where f and h are non-negative, f is finite a.e. and , h are Lebesgue integrable, has an absolutely continuous solution.     

A new augmented Lyapunov-Krasovskii functional approach for stability of linear systems with time-varying delays

This paper proposes improved delay-dependent conditions for asymptotic stability of linear systems with time-varying delays. The proposed method employs a suitable Lyapunov-Krasovskii’s functional for new augmented system. Based on Lyapunov method, delay-dependent stability criteria for the systems are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various optimization algorithms. Three numerical examples are included to show that the proposed method is effective and can provide less conservative results.     

Periodic solutions and stability for a singularly perturbed linear delay differential equation

We describe the bifurcation hypersurfaces for periodic solutions of a singularly perturbed linear differential difference equation in the space of the coefficients of the equation. For low dimension we show that the locus of stability of that equation approaches the locus of stability of a limit difference equation.     

Attracting and invariant sets for a class of impulsive functional differential equations

In this article, a class of nonlinear and nonautonomous functional differential systems with impulsive effects is considered. By developing a delay differential inequality, we obtain the attracting set and invariant set of the impulsive system. An example is given to illustrate the theory.     

Lyapunov-type inequalities for a class of higher-order linear differential equations

In this paper, we obtain some Lyapunov-type inequalities for a class of higher-order linear differential equations. The results of this paper generalize and improve some earlier results on this topic.     

Positive solutions for a scalar differential equation with several delays

For a scalar delay differential equation , we obtain new explicit conditions for the existence of a positive solution.     

On the stability and boundedness of solutions of a kind of third order delay differential equations

In this paper, we study Eq. (1.1) for asymptotic stability of the zero solution when and uniformly bounded and uniformly ultimate bounded of all solutions when      

Elementary remarks on Ulam-Hyers stability of linear functional equations

In this paper the general method for proving stability of linear functional equations is described.     

On almost periodic mild solutions for stochastic functional differential equations

In this paper, a class of stochastic functional differential equations given by

     

Solutions with exponential character to a linear functional equation and roots of its characteristic equation

We study the connection between solutions of the functional equation φ(x)=S_∫φ(x+M(s))σ(ds) which are comparable at infinity with functions of the form xeλx, and real roots of the characteristic equation S_∫eλM(s)σ(ds)=1.     

Existence and stability in the α-norm for partial functional differential equations of neutral type

In this work, we study a general class of partial neutral functional differential equations. We assume that the linear part generates an analytic semigroup and the nonlinear part is Lipschitz continuous with respect to the é-norm associated to the linear part. We discuss the existence, uniqueness, regularity and stability of solutions. Our results are illustrated by an example. This work extends previous results on partial functional differential equations (Fitzgibbon and Parrot, Nonlinear Anal., TMA 16, 479–487 (1991), Hale, Rev. Roum. Math. Pures Appl. 39, 339–344 (1994), Hale, Resen. Inst. Mat. Estat. Univ. Sao Paulo 1, 441–457 (1994), Travis and Webb, Trans. Am. Math. Soc. 240 129–143 (1978), Wu and Xia, J. Differ. Equ. 124 247–278 (1996)). Mathematics Subject Classification (1991) 34K20, 34K30, 34K40, 47D06     

Generalized Halanay inequalities for dissipativity of Volterra functional differential equations

This paper is concerned with the dissipativity of theoretical solutions to nonlinear Volterra functional differential equations (VFDEs). At first, we give some generalizations of Halanay's inequality which play an important role in study of dissipativity and stability of differential equations. Then, by applying the generalization of Halanay's inequality, the dissipativity results of VFDEs are obtained, which provides unified theoretical foundation for the dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay-integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice.     

Exponential stability of impulsive stochastic functional differential equations

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.     

Solutions with constant sign at infinity of a linear functional equation of infinite order

We investigate how the existence and behaviour of solutions φ with a constant sign of the equation

     

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.     

Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation

The paper deals with convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. It is proved that the semi-implicit Euler method is convergent with strong order . The conditions under which the method is MS-stable and GMS-stable are determined and the numerical experiments are given.     

Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations

This paper studies the practical stability of the solutions of nonlinear impulsive functional differential equations. The obtained results are based on the method of vector Lyapunov functions and on differential inequalities for piecewise continuous functions. Examples are given to illustrate our results.     

Linear functional differential equations with Property A

In the paper the general linear functional differential equation with several distributed deviations is considered. Sufficient conditions for the equation to have Property A (see Definition 1.2 below) are established. The obtained results are new even for Eq. (1.4).     

The exact boundaries of the stability domains of linear differential equations with distributed delay

For a differential equation with a distributed varying delay, sufficient criteria for the asymptotic and uniform stability of solutions are obtained. The constructed examples demonstrate exactness of the boundary of the obtained stability domain.