Stability Of Delay Differential Equations Research Articles

An improvement in stability of delay difference systems

A new result on the stability of delay difference equations is established, in which a kind of weighted norm forx is adopted in order to estimate the upper bound of Liapunov functional.

Uniform Asymptotic Stability of Delay Differential Equations with Amnesia

Uniform Asymptotic Stability of Delay Differential Equations with Amnesia

Ultimate Bounds and Global Asymptotic Stability for Differential Delay Equations

Ultimate Bounds and Global Asymptotic Stability for Differential Delay Equations

A Velocity Functional for an Analysis of Stability in Delay-Differential Equations

The author considers the linear delay-differential system (*)$\dot x(t) = A_0 x(t) + \sum\nolimits_1^p {A_i } x(t - h_i )$. It is shown that there is a velocity functional which along with its Lie derivative is analogous in the theory of delay-differential operators to the velocity Lyapunov function $V(x) = \langle {\langle {Ax,Ax} \rangle } \rangle $, which along with its Lie derivative $x^T (A + A^T )x$ is used in the analysis of the ordinary differential equation $\dot x(t) = Ax(t)$. The Lie derivative can be written as $2\langle {(A_0 + \sum\nolimits_1^p {\sigma _i } A_i )\phi ,\phi } \rangle $ in a suitable inner product $\langle , \rangle $ for $C^1 $ vector functions $\phi $ given over $[ - \eta ,0]$, where the $\sigma _i $ signify delay operators having length $h_i $ and $\eta = \max (h_1 , \ldots ,h_p )$. Next considering the nonlinear delay-differential equation (†)$\dot x(t) = f(x(t),x(t - h_1 ), \ldots ,x(t - h_p ))$, the author gives a velocity functional having Lie derivative which locally resembles that for the linear system (*). It is proven that if the linear operator associated with this Lie derivative everywhere satisfies a certain stability property, then the nonlinear system (†) will be globally contractive to a unique equilibrium.

Intrinsic parameters and stability of differential-delay equations

Through the efforts of many authors, analytic solutions and explicit stability criteria have been developed for many types of linear differentialdelay equations. The treatise of Bellman and Cooke [3] provides a good account of work done before 1963; active research in the field continues up to the present. Besides the concrete results obtained for specific classes of equations, there are also some rough “general principles” that have long guided users of differential-delay equations in their efforts to understand the effects of the various kinds of time delays on stability. One such principle says that instability will be present when the ratio of the time delay to the relaxation time of the system is large. However, the “stability switching” phenomenon, discovered as early as 1948 by Ansoff and Krumhansl [2], contradicts this principle, and a great variety of responses to the presence of time lags are now known. In their landmark paper on the subject, Cooke and Grossman [4] included a series of key examples illustrating the things that can happen. It seems unlikely that any simple general principles will be found to serve as reliable indicators of stability. Instead, we can reasonably hope that the development of further principles of restricted applicability will gradually fill in more of the overall picture. To this end, we introduce a new framework for the classification and representation of differential-delay equations (or “DDE?‘) based on the simplest notions of probability theory. Distributed time delays are represented as probability measures whose expectations and variances can be used to construct intrinsic parameters for the classes of DDEs studied. Intrinsic parametrization has two advantages.

Interaction between oscillations and global asymptotic stability in delay differential equations

Interaction between oscillations and global asymptotic stability in delay differential equations

Mean-square stability of delay-differential equations with non-stationary random coefficients

Abstract The mean-square asymptotic stability of the equilibrium stale of a system of linear stochastic delay-differential equations with general non-stationary random coefficients is studied. The Lyapunov method of stability analysis is employed and several criteria for the mean-square stability of the equilibrium state of the system are established. A few examples are solved and the application of the method is discussed.

On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations

Asymptotic stability of delay differential equations of the retarded type is studied in terms of zero criteria for polynomials in two independent complex variables. The major new result of this note is that asymptotic stability independent of delay can be expressed in terms of a (finitely) implementable algebraic criterion involving a two-variable polynomial.