Asymptotic stability of general linear methods for systems of linear neutral delay differential-algebraic equations

This paper is concerned with numerical stability of general linear methods (GLMs) for a system of linear neutral delay differential-algebraic equations. A sufficient and necessary condition for asymptotic stability of GLMs solving such system is derived. Based on this main result, we further investigate the asymptotic stability of linear multistep methods, Runge–Kutta methods, and block θ-methods, respectively. Numerical experiments confirm our theoretical result.

[1]  L. Shampine,et al.  Delay-differential-algebraic equations in control theory , 2006 .

[2]  Stephen L. Campbell,et al.  Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions , 2009, Appl. Math. Comput..

[3]  Uri M. Ascher,et al.  The numerical solution of delay-differential-algebraic equations of retarded and neutral type , 1995 .

[4]  Hongjiong Tian,et al.  The asymptotic stability of one-parameter methods for neutral differential equations , 1994 .

[5]  Ernst Hairer,et al.  Computing breaking points in implicit delay differential equations , 2008, Adv. Comput. Math..

[6]  Lin,et al.  THE STABILITY OF LINEAR MULTISTEP METHODS FOR LINEAR SYSTEMS OF NEUTRAL DIFFERENTIAL EQUATIONS , 2001 .

[7]  Linda R. Petzold,et al.  Asymptotic stability of linear delay differential-algebraic equations and numerical methods , 1997 .

[8]  Hongjiong Tian,et al.  Asymptotic Stability of Linear Neutral Delay Differential-Algebraic Equations and Linear Multistep Methods , 2011, SIAM J. Numer. Anal..

[9]  L. Vlasenko.  Existence and uniqueness theorems for an implicit delay differential equation , 2000 .

[10]  Hongjiong Tian,et al.  Asymptotic Stability of Linear Neutral Delay Differential-Algebraic Equations and Runge-Kutta Methods , 2014, SIAM J. Numer. Anal..

[11]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[12]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[13]  Qianshun Chang,et al.  Linear stability of general linear methods for systems of neutral delay differential equations , 2001, Appl. Math. Lett..

[14]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[15]  T. Mitsui,et al.  Asymptotic and numerical stability of systems of neutral differential equations with many delays , 2009 .

[16]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[17]  Hao Chen,et al.  Stability analysis of linear multistep and Runge-Kutta methods for neutral multidelay-differential-algebraic systems , 2012, Math. Comput. Model..

[18]  R. Hauber,et al.  Numerical treatment of retarded differential–algebraic equations by collocation methods , 1997, Adv. Comput. Math..

[19]  S. Campbell.  Singular linear systems of differential equations with delays , 1980 .

[20]  Chengjian Zhang.  NGP(?)-stability of general linear methods for NDDEs*1, *2 , 2004 .

[21]  M. N. Spijker,et al.  Stability analysis of numerical methods for delay differential equations , 1991 .

[22]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[23]  Baruch Cahlon,et al.  Algebraic criteria for stability of linear neutral systems with a single delay , 2001 .

[24]  Pierdomenico Pepe,et al.  On the asymptotic stability of coupled delay differential and continuous time difference equations , 2005, Autom..