Stability properties of differential-algebraic equations and Spin-stabilized discretizations.

Classical stability properties of solutions that are well-known for ordinary differential equations (ODEs) are generalized to differential-algebraic equations (DAEs). A new test equation is derived for the analysis of numerical methods applied to DAEs with respect to the stability of the numerical approximations. Morevover, a stabilization technique is developed to improve the stability of classical DAE integration methods. The stability regions for these stabilized discretization methods are determined and it is shown that they much better reproduce the stability properties known for the ODE case than in the unstabilized form. Movies that depict the stability regions for several methods are included for interactive use.

[1]  Johannes P. Schlöder,et al.  Real-Time Optimization for Large Scale Processes: Nonlinear Model Predictive Control of a High Purity Distillation Column , 2001 .

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

[3]  Roswitha März,et al.  Analyzing the stability behaviour of solutions and their approximations in case of index-2 differential-algebraic systems , 2002, Math. Comput..

[4]  Ricardo Riaza.  Stability Issues in Regular and Noncritical Singular DAEs , 2002 .

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

[6]  Tatjana Stykel,et al.  Stability and inertia theorems for generalized Lyapunov equations , 2002 .

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

[8]  Tatjana Stykel,et al.  On criteria for asymptotic stability of differential-algebraic equations , 2002 .

[9]  Inmaculada Higueras,et al.  Stability preserving integration of index-1 DAEs , 2003 .

[10]  Peter Kunkel,et al.  Symmetric collocation methods for linear differential-algebraic boundary value problems , 2002, Numerische Mathematik.

[11]  C. W. Gear,et al.  The index of general nonlinear DAEs , 1995 .

[12]  Uri M. Ascher,et al.  Stability of Computational Methods for Constrained Dynamics Systems , 1991, SIAM J. Sci. Comput..

[13]  R. Russell,et al.  Unitary integrators and applications to continuous orthonormalization techniques , 1994 .

[14]  Diederich Hinrichsen,et al.  Mathematical Systems Theory I , 2006, IEEE Transactions on Automatic Control.

[15]  Volker Mehrmann,et al.  Regular solutions of nonlinear differential-algebraic equations and their numerical determination , 1998 .

[16]  K. Wright.  Differential equations for the analytic singular value decomposition of a matrix , 1992 .

[17]  Volker Mehrmann,et al.  Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index , 2004, Numerische Mathematik.

[18]  B. Ingalls.  Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness [Book Review] , 2006, IEEE Control Systems.

[19]  M. Diehl,et al.  Optimization of multiple‐fraction batch distillation with recycled waste cuts , 2002 .

[20]  Michael Günther,et al.  CAD based electric circuit modeling in industry. Pt. 2: Impact of circuit configurations and parameters , 1997 .

[21]  R. Glowinski,et al.  Numerical methods for multibody systems , 1994 .

[22]  R. März,et al.  Stability preserving integration of index-2 DAEs , 2003 .

[23]  A. Bunse-Gerstner,et al.  Numerical computation of an analytic singular value decomposition of a matrix valued function , 1991 .

[24]  V. Mehrmann,et al.  Smooth factorizations of matrix valued functions and their derivatives , 1991 .

[25]  Werner C. Rheinboldt,et al.  CLASSICAL AND GENERALIZED SOLUTIONS OF TIME-DEPENDENT LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS , 1996 .

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

[27]  Stephen L. Campbell,et al.  Comment on controlling generalized state-space (descriptor) systems , 1987 .

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

[29]  T. Stykel.  Analysis and Numerical Solution of Generalized Lyapunov Equations , 2002 .

[30]  Volker Mehrmann,et al.  Analysis of Over- and Underdetermined Nonlinear Differential-Algebraic Systems with Application to Nonlinear Control Problems , 2001, Math. Control. Signals Syst..

[31]  W. Rheinboldt.  Differential-algebraic systems as differential equations on manifolds , 1984 .

[32]  A. Bountis.  Dynamical Systems And Numerical Analysis , 1997, IEEE Computational Science and Engineering.

[33]  Roswitha März,et al.  Solvability of linear differential algebraic equations with properly stated leading terms , 2004 .

[34]  W. Rheinboldt,et al.  Theoretical and numerical analysis of differential-algebraic equations , 2002 .

[35]  Michael Günther,et al.  CAD based electric circuit modeling in industry. Pt. 1: Mathematical structure and index of network equations , 1997 .

[36]  Ebroul Izquierdo Macana,et al.  On Asymptotics in Case of Linear Index-2 Differential-Algebraic Equations , 1998 .

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

[38]  W. R. Howard,et al.  Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness , 2005 .

[39]  W. Rheinboldt.  On the computation of multi-dimensional solution manifolds of parametrized equations , 1988 .