Stability Of Delay Equations Research Articles

Stability of delay equations

For a large class of nonautonomous linear delay equations with distributed delay, we obtain the equivalence of hyperbolicity, with the existence of an exponential dichotomy, and Ulam–Hyers stability. In particular, for linear equations with constant or periodic coefficients and with a simple spectrum these two properties are equivalent. We also show that any linear delay equation with an exponential dichotomy and its sufficiently small Lipschitz perturbations are Ulam–Hyers stable.

On Stability of Delay Equations with Positive and Negative Coefficients with Applications

We obtain new explicit exponential stability conditions for linear scalar equations with positive and negative delayed terms \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))- \sum_{k=1}^l b_k(t)x(g_k(t))=0 and its modifications, and apply them to investigate local stability of Mackey–Glass type models \dot{x}(t)=r(t)\left[\beta\frac{x(g(t))}{1+x^n(g(t))}-\gamma x(h(t))\right] and \dot{x}(t)=r(t)\left[\beta\frac{x(g(t))}{1+x^n(h(t))}-\gamma x(t)\right],

Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation can be oscillating and asymptotically unstable, the delay equation , where , can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion. MSC:34K20.

On some constants for oscillation and stability of delay equations

We discuss the famous constants of1/e1/e, 1,3/23/2in necessary and/or sufficient oscillation and stability conditions for delay differential equations with one or more delays:\[x′(t)=−∑k=1mak(t)x(t−hk(t)),x^{\prime }(t)= - \sum _{k=1}^ma_k(t) x(t-h_k(t)),\]including equations with oscillatory coefficients. Some counterexamples (which refer to necessary oscillation and stability conditions) are presented and open problems are stated.

Stability Theory of General Contractions for Delay Equations

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v′ = L(t)vt with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))vt, thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v′ = L(t)vt + f(t, vt). In addition, we consider general contractions e−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.

Stability of delay equations via Lyapunov functions

The importance of Lyapunov functions is well known. In the general setting of nonautonomous linear delay equations v ′ = L ( t ) v t , we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of Lyapunov functions. This includes uniform exponential behavior as a very special case, and it provides an alternative (usually simpler and particularly more direct) approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations.

Inverse Routh table construction and stability of delay equations

Based on an inversion of the Routh table construction, a unimodular characterization of all Hurwitz polynomials is obtained. In parameter space, the Hurwitz polynomials of degree n correspond to the positive 2 n - tant . The method is then used to construct classes of stable continuous-time delay-difference equations and delay differential equations of neutral type, by a suitable limiting process.

Short proofs and a counterexample for analytical and numerical stability of delay equations with infinite memory

In this paper we study a class of linear systems of delay differential equations with variable coefficients and variable delay with infinite memory. This kind of problem includes the well-known class of equations with proportional delay (the pantograph equations). The aims of this paper are those of investigating the asymptotic behaviour of both the analytical and the numerical solutions, which are obtained when suitable discrete methods are applied. Relevant to the constant-coefficient equations with proportional delay, we first give an alternative short proof of an important result due to Iserles concerning sufficient conditions for the asymptotic stability of the solutions. Then we establish a new stability result for the more general case, which improves the conditions known in the literature. Afterwards, we focus our attention on the behaviour of one-leg Θ-methods implemented on special integration meshes and prove a numerical stability result under suitable assumptions on the coefficient matrices. Doing this, we note that the numerical stability result applies both to the constant-coefficient and to the general variable-coefficient case under formally analogous conditions. Nevertheless, we prove that, in the general variable-coefficient framework, such conditions do not imply asymptotic stability of the true solutions. This is proved by constructing an explicit counterexample.

Positivity and stability of delay equations with nonautonomous past

AbstractIn our paper we discuss the asymptotic behaviour of a linear delay equation with nonautonomous past. Using positivity and compactness assumptions we obtain a quite complete characterization of the asymptotic behaviour. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Absolute stability in delay equations

An equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays. This is equivalent to a certain characteristic equation's having all its roots in the left half-plane for all delays. We obtain a simple necessary and sufficient condition for absolute stability of a class of delay equations.