On criteria for asymptotic stability of differential-algebraic equations

This paper discusses Lyapunov stability of the trivial solution of linear differential-algebraic equations. As a criterion for the asymptotic stability we propose numerical parameters characterizing the property of a regular matrix pencil λA - B to have all finite eigenvalues in the open left half-plane. Numerical aspects for computing these parameters are discussed.

[1]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[2]  An effectively calculable parameter for the stability property of a system of linear differential equations with constant coefficients , 1980 .

[3]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[4]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[5]  J. L. V. Dorsselaer,et al.  Pseudospectra for matrix pencils and stability of equilibria , 1997 .

[6]  Peter Lancaster,et al.  ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS , 1966 .

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

[8]  Caren Tischendorf.  On the stability of solutions of autonomous index-I tractable and quasilinear index-2 tractable DAEs , 1994 .

[9]  G. Hewer,et al.  The sensitivity of the stable Lyapunov equation , 1987, 26th IEEE Conference on Decision and Control.

[10]  Edmond A. Jonckheere,et al.  New bound on the sensitivity of the solution of the Lyapunov equation , 1984 .

[11]  Thilo Penzl,et al.  Numerical solution of generalized Lyapunov equations , 1998, Adv. Comput. Math..

[12]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[13]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[14]  S. Godunov.  Problem of the dichotomy of the spectrum of a matrix , 1986 .

[15]  Alan J. Laub,et al.  Algorithm 705; a FORTRAN-77 software package for solving the Sylvester matrix equation AXBT + CXDT = E , 1992, TOMS.

[16]  Roswitha März,et al.  Criteria for the Trivial Solution of Differential Algebraic Equations with Small Nonlinearities to be Asymptotically Stable , 1998 .

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

[18]  W. Rudin.  Real and complex analysis , 1968 .

[19]  Enrique S. Quintana-Ortí,et al.  Solving stable generalized Lyapunov equations with the matrix sign function , 1999, Numerical Algorithms.

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

[21]  Nancy Nichols,et al.  On the stability radius of a generalized state-space system , 1993 .

[22]  J. Demmel,et al.  An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems , 1997 .

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

[24]  Peter C. Müller,et al.  Stability of Linear Mechanical Systems With Holonomic Constraints , 1993 .

[25]  S. Godunov.  Modern Aspects of Linear Algebra , 1998 .

[26]  S. Campbell.  Singular Systems of Differential Equations , 1980 .

[27]  Alan J. Laub,et al.  Solution of the Sylvester matrix equation AXBT + CXDT = E , 1992, TOMS.

[28]  S. Hammarling.  Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .