Analysis of Linear Variable Coefficient Delay Differential-Algebraic Equations

The analysis of general linear variable coefficient delay differential-algebraic systems (DDAEs) is presented. The solvability for DDAEs is investigated and a reformulation procedure to regularize a given DDAE is developed. Based on this regularization procedure existence and uniqueness of solutions and consistency of initial functions is analyzed as well as other structural properties of DDAEs like smoothness requirements. We also present some examples to demonstrate that for the numerical solution of a DDAE, a reformulation of the system before applying numerical methods is essential.

[1]  Ricardo Riaza,et al.  Differential-Algebraic Systems: Analytical Aspects and Circuit Applications , 2008 .

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

[3]  Navier–Stokes equations with delays on unbounded domains , 2006 .

[4]  C. Baker,et al.  Differential algebraic equations with after-effect , 2002 .

[5]  S. Niculescu,et al.  Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach , 2007 .

[6]  Frank Kelly,et al.  Mathematical Modelling of the Internet , 2001 .

[7]  A. Steinbrecher.  Regularization of Quasi-Linear Differential-Algebraic Equations , 2015 .

[8]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[9]  Donald F. Towsley,et al.  Analysis and design of controllers for AQM routers supporting TCP flows , 2002, IEEE Trans. Autom. Control..

[10]  B. Engquist,et al.  Mathematics Unlimited: 2001 and Beyond , 2001 .

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

[12]  Stephen L. Campbell.  Nonregular 2D descriptor delay systems , 1995 .

[13]  Volker Mehrmann,et al.  ANALYSIS AND REFORMULATION OF LINEAR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS ‡ , 2012 .

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

[15]  E. Griepentrog,et al.  Differential-algebraic equations and their numerical treatment , 1986 .

[16]  Albert E. Ruehli,et al.  Retarded models for PC board interconnects-or how the speed of light affects your SPICE circuit simulation , 1991, 1991 IEEE International Conference on Computer-Aided Design Digest of Technical Papers.

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

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

[19]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[20]  T. Caraballo,et al.  Navier-Stokes equations with delays , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[21]  M. A. Akanbi,et al.  Numerical solution of initial value problems in differential - algebraic equations , 2005 .

[22]  Qing-Chang Zhong,et al.  Robust Control of Time-delay Systems , 2006 .

[23]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[24]  Martin Stynes,et al.  Numerical Treatment of Partial Differential Equations , 2007 .

[25]  Weijiu Liu,et al.  ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF TIME-DELAYED BURGERS' EQUATION , 2001 .

[26]  Vishal Misra,et al.  Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED , 2000, SIGCOMM.

[27]  Charles William Gear.  Simulation: Conflicts between Real-Time and Software , 1977 .

[28]  Volker Mehrmann,et al.  Transformation of high order linear differential-algebraic systems to first order , 2006, Numerical Algorithms.

[29]  Timo Eirola,et al.  On Smooth Decompositions of Matrices , 1999, SIAM J. Matrix Anal. Appl..

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

[31]  钟 庆昌.  Robust Control of Time-delay Systems , 2006 .