An Analysis of Delay-Dependent Stability for Ordinary and Partial Differential Equations with Fixed and Distributed Delays

Abstract

This paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays, and with the study of the stability of discretizations of such differential equations. We start with a delay-dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients. We study the exact stability region of the continuous problem as a function of the parameters of the model. Next, it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems. In the second part of the paper, we study delay partial differential equations. The stability region of the fully continuous problem is analyzed first. Then a semidiscretization in space is applied. It is shown that the spatial discretization leads to a reduction of the stability region when the standard second-order central difference operator is employed to approximate the diffusion operator. Finally we consider the delay-dependent stability of the fully discrete problem, where the partial differential equation is discretized both in space and in time.Some numerical examples and further discussions are given.

Stability of the heat and of the wave equations with boundary time-varying delays

“…The stability of linear parabolic systems with constant coefficients and internal constant delays has been studied in [6] in the frequency domain.…”

Delay-dependent stability of high order Runge–Kutta methods

“…In particular, they developed in [13] the relationship between stability analysis and order stars, which leads to an elegant result that all Gauss methods are τ (0)-stable. For the delay-dependent stability analysis for complex coefficient equations or other type equations, we refer the reader to [12,17,[19][20][21]. We remark that the recent studies in [12,20] show that no Runge-Kutta method can completely preserve the asymptotic stability of general multidimensional delay systems.…”

“…This is because they behave very well in practice so that it is valuable to provide a firmer and more detailed theoretical support for them. In addition, the large system obtained from the spatial discretization for a partial differential equation with delays possesses a special construction [17]. To this class of systems, the delay-dependent stability results on real coefficient equations are directly applicable.…”

Delay-dependent stability of numerical methods for delay differential systems of neutral type

“…Although numerical delay-independent stability has been discussed in [1,2,10], only a few works have been reported for the delay-dependent case [2,7,13,18]. The literature [2,7,18] proposed the delay-dependent stability of numerical methods for the systemu (t) = Lu(t) + Mu(t − τ ), (1.3) and called it as D-stability.…”