Asymptotic Stability of Linear Neutral Delay Differential-Algebraic Equations and Linear Multistep Methods

Abstract

This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result.

Spectrum-Based Robust Stability Analysis of Linear Delay Differential-Algebraic Equations

“…Otherwise, the exponential stability of (19.1) is said to be delay-dependent. The following sufficient criterion was established in [13], which improved the one earlier given in [67], and later it was further developed in [63]. The following theorem for the delay-independent exponential stability of (19.1) is a special case of Theorem 1 in [13].…”

“…Proposition 3.4 in [18] shows that the allowable delay conditions ND 21 D 0 and ND 22 D 0 are simultaneously satisfied if and only if We note that although the spectral criterion˛.H / < 0 was used for characterizing the exponential stability of DDAEs with regular pair .E; A/ of arbitrary index in [13,63,67], a rigorous proof had not been available prior to the appearance [18].…”

“…Finally, we emphasize a numerical consequence of delay-independent exponential stability. In [13,63,67], it is shown that for delay-independently exponentially stable DDAEs, numerical solutions by certain classes of A-stable discretization methods preserve the exponential stability.…”

“…Many of the results that are known for DODEs cannot be extended to the DDAE case. Even the well-known spectral analysis for the exponential stability or the asymptotic stability of linear timeinvariant DDAEs (19.1) is much more complex than that for DAEs and DDEs, see [13,63,67] for some special cases.…”

Stability and Robust Stability of Linear Time-Invariant Delay Differential-Algebraic Equations

“…Many of the results that are known for DDEs do not carry over to the DDAE case. Even the well-known spectral analysis for the exponential stability or the asymptotic stability of linear time-invariant DDAEs (1) is much more complex than that for DAEs and DDEs, see [10,39,43] for some special cases.…”