Linear Delay Equations Research Articles

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 p= 1 2 . The conditions under which the method is MS-stable and GMS-stable are determined and the numerical experiments are given.

The Modulus Semigroup for Linear Delay Equations

The main purpose of this paper is describing the generator of the modulus semigroup of the C0-semigroup associated with the delay equation ( u ' (t) = Au(t) + Lut (t ≥ 0) , u(0) = x ∈ X, u0 = f ∈ Lp(−h,0; X) , in the Banach lattice X × Lp(−h,0; X), where X is a Banach lattice with order continuous norm. As a preparation it is shown that W 1 p(a, b; X) is a sublattice of Lp(a, b; X), for 1 ≤ p < ∞. A further preparation is the computation of the modulus of the operator L appearing above. Also, we establish a result concerning the existence of the modulus semigroup for C0- semigroups acting in KB-spaces.

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.

Parameter and initial function identifiabilty of linear delay differential equations

Parameter identifiability is studied for a class of finite and infinite dimensional systems described by convolution equations. For linear differential delay equations of neutral type and with distributed delays, it is shown how the identifiability of the plant can be formulated in terms of controllability conditions, namely approximate controllability. Identifiability of the initial condition, as well as the notion of sufficiently rich input which enforces identifiability are also addressed, and the results are obtained assuming knowledge of the solution on a bounded time interval.

Identifiability analysis of a class of systems described by convolution equations

Parameter identifiability is studied for a class of finite and infinite dimensional systems described by convolution equations. For linear differential delay equations of neutral type and with distributed delays, it is shown how the identifiability property can be formulated in terms of controllability conditions, namely approximate controllability. The notion of sufficiently rich input which enforces identifiability is also adressed, and the results are obtained assuming knowledge of the solution on a bounded time interval.

Parameter identifiability of differential delay equations

AbstractParameter identifiability is concerned with the question whether the parameters of a specific model can be identified from knowledge about certain solutions of the model, assuming perfect data. In this paper we shall consider conditions for identifiability of parameters and time delays in infinite dimensional dynamical systems, described by linear differential delay equations, assuming knowledge of particular solutions on bounded time intervals. We shall show how methods from operator theory can be exploited to give necessary and sufficient conditions which guarantee that the inverse problem has a unique solution. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Strong discrete time approximation of stochastic differential equations with time delay

The paper introduces an approach for the derivation of discrete time approximations for solutions of stochastic differential equations (SDEs) with time delay. The suggested approximations converge in a strong sense. Furthermore, explicit solutions for linear stochastic delay equations are given.

Robust stability of uncertain stochastic differential delay equations

In this paper we first discuss the robust stability of uncertain linear stochastic differential delay equations. We then extend the theory to cope with the robust stability of uncertain semi-linear stochastic differential delay equations. We shall also give several examples to illustrate our theory.

Existence of Nonoscillatory Solution of Second Order Linear Neutral Delay Equation

Consider the neutral delay differential equation with positive and negative coefficients,[formula]wherep∈Rand[formula]Some sufficient conditions for the existence of a nonoscillatory solution of the above equation expressed in terms of ∫∞sQi(s)ds<∞,i=1,2, and certain technical conditions implying thatQ1(s) dominatesQ2(s) are obtained for values ofp≠±1.

Preservation of Stability in Delay Equations under Delay Perturbations

We consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations. As an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays. We also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays.

On the sign definiteness of Liapunov functionals and stability of a linear delay equation

In this paper we give a new denition of the positive-deniteness of the Liapunov functional involved in the stability and asymptotic stability investigation. Using this notion we prove Liapunov type theorems and apply these results to the scalar equation _ x(t) = b(t)x(t r(t)), where b(t) and r(t) may be unbounded.

Semi-Hyperbolic Mappings, Condensing Operators, and Neutral Delay Equations

Semi-hyperbolic mappings in Banach spaces are Lipschitz continuous and not necessarily invertible. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. It is shown that semi-hyperbolic mappings are locallyψ-contracting, whereψis the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it isψ-contracting and has no spectral values on the unit circle. A bishadowing result, which combines both direct and indirect forms of shadowing, is extended to semi-hyperbolic mappings in Banach spaces with locally condensing continuous comparison mappings. The result is applied to linear neutral delay equations with nonsmooth perturbations.

On the Sharpness of a Theorem by Cooke and Verduyn Lunel

The sharpness of a theorem of K. Cooke and S. M. Verduyn Lunel (Differential Integral Equations6(1993) 1101–1117) about small solutions of linear time-dependent delay equations is proved. A wide class of nonautonomous functional-differential equations possessing small solutions is constructed. The construction is based on the existence of quickly decreasing solutions of functional-differential equations with linearly expanded arguments.

Bi-shadowing and delay equations

Bi-shadowing is an extension to the concept of shadowing and is usually used in the context of comparing computed trajectories with the true trajectories of a dynamical system in Rn. Here the concept is defined in a Banach space and is applied to delay equations to give an apparently new result on nonlinear perturbations of linear delay equations. This is essentially a form of robustness with respect to small nonlinear disturbances.

Solution moment stability in stochastic differential delay equations.

We study the behavior of the first and second solution moments for linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. In the presence of additive noise (white or colored), the stability domain of both moments is identical to that of the unperturbed system. When these moments lose stability, there is a Hopf bifurcation and the first moment oscillates with a period identical to the solution of the unperturbed equation, while the oscillation period of the second moment is exactly one half the period of the unperturbed solution and the first moment. When perturbations are of the parametric (or multiplicative) type and white noise is assumed, under the It\^o interpretation the first moment of the solution preserves properties of the solution of the deterministic equation, while the behavior of the second moment depends on the amplitude of the stochastic perturbation. The critical delay value at which the second moment loses stability and becomes oscillating is derived, and it is less than the critical delay for the first moment. Under the Stratonovich interpretation, quite different properties were observed for the moment equations, namely, various critical values of the delay and period of oscillations. For the case of parametric colored noise perturbations, sufficient (p-stability) conditions are derived which are independent of the value of delay, and it is shown that colored noise has a stabilizing effect with respect to white noise.

L2-Perturbation of a Linear Delay Differential Equation

In this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation ẋ (t) = ( a 0 + a( t)) x( t) + ( b 0 + b( t)) x( t − τ), assuming that it is an L 2-perturbation of the autonomous equation ẋ( t) = a 0 x( t) + b 0 x( t − τ). Our theorem is an essential generalization of the corresponding result in the classical book by R. Bellman and K. L. Cooke. The proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation.

A Conjecture on the Density of a System of Weighted Polynomial Functions

A conjecture on the density of a system of weighted polynominal functions is proposed. This system of weighted polynominal functions arises in the discussion of linear differential delay equations. By the known results for the linear differential delay equations, we confirmed the conjecture in some special cases. This paper attempts to draw the attention of people working in analysis and approximation theory.

Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments

The idea of approximating the solutions of delay differential equations by equations with piecewise constant arguments (EPCA) has been suggested by Gyijri [l], who proved convergence of the method for linear and nonlinear delay equations on compact intervals, and under certain conditions also on the half line. The approximating equations with piecewise constant argument can, in turn, be solved by use of difference equations. The latter then also provide an approximation of the original delay equation. (The theory of EPCA was initiated and studied by Cooke and Wiener in [2] and [3].) In this paper, we begin to investigate the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved. A number of authors have dealt with the problem of showing that discretization of ordinary differential equations does not significantly alter the basic qualitative features. Readers may refer to the papers of Kloeden and Lorenz [4] and Beyn [5] for some of this work. For delay differential equations, questions of this sort have been studied by Cryer [6], Barwell [7], Zennaro [8], and later authors. The paper of Strehmel et al. [9] contains an up-to-date list of references. In our method of approximation, the delay equation is first replaced by an EPCA, and then by a difference equation, and our objective is to relate the qualitative dynamics of these three equations. We obtain results for autonomous and nonautonomous equations with one or many delays. For autonomous equations, the resulting difference algorithm is just a simple Euler scheme, but for nonautonomous equations, it includes other possibilities. In order to make the method and the nature of results as clear as possible, we begin with a particular test equation, k(t) = -pz(t 7), p > 0, 7 > 0.

Noise and stability in differential delay equations

We study the stability of linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. Using a stochastic analog of the second Liapunov method, sufficient conditions for mean square and stochastic stability are derived.

A method for designing a stabilizing control for a class of uncertain linear delay systems

We present a simple method for designing stabilizing controllers for linear delay equations containing unmatched uncertainty. The methodology employed is based on differential inequalities rather than on Lyapunov stability theory.