Criteria for the Trivial Solution of Differential Algebraic Equations with Small Nonlinearities to be Asymptotically Stable

März, Roswitha

doi:10.1006/jmaa.1998.6055

Cited by 25 publications

(20 citation statements)

References 14 publications

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“…That is why most stability results in the literature are obtained for DAEs of index 1, see [1,[6][7][8]11,13,15,23,24,27,30,32,34]. Stability results for higher-index DAEs exist only in the case if special structured problems are considered and(or) extra assumptions are necessary [28,32,35,36]. Another alternative way is to reformulate the DAE by applying some index reduction technique in order to obtain lower-index DAEs which possess the same solution set, e.g.…”

Section: Remark 32mentioning

confidence: 98%

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On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems

Chyan

Linh

2008

Journal of Differential Equations

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This paper is addressed to some questions concerning the exponential stability and its robustness measure for linear timevarying differential-algebraic systems of index 1. First, the Bohl exponent theory that is well known for ordinary differential equations is extended to differential-algebraic equations. Then, it is investigated that how the Bohl exponent and the stability radii with respect to dynamic perturbations for a differentialalgebraic system depend on the system data. The paper can be considered as a continued and complementary part to a recent paper on stability radii for time-varying differential-algebraic equations [N.H. Du, V.H. Linh, Stability radii for linear timevarying differential-algebraic equations with respect to dynamic perturbations, J. Differential Equations 230 (2006) 579-599].

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Section: Remark 32mentioning

confidence: 98%

“…Note that the index of the linearized DAE system may depend on the solution in consideration, as well. We refer to [1,7,8,13,15,23,24,27,28,31,32,[34][35][36] for some recent stability results for DAEs and their numerical solutions.…”

mentioning

confidence: 99%

On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems

Chyan

Linh

2008

Journal of Differential Equations

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“…Different stability concepts for DAEs have been discussed already in [2,42,43,53,60,62,68,69,71,72,73,74]. Only very few papers, however, discuss the spectral theory for DAEs, see [17,18] for results on Lyapunov exponents and Lyapunov regularity, [57] for the concept of exponential dichotomy used in numerical solution to boundary value problems, and [16,35] for robustness results of exponential stability and Bohl exponents.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Linh

Mehrmann

2009

J Dyn Diff Equat

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Lyapunov and exponential dichotomy spectral theory is extended from ordinary differential equations (ODEs) to nonautonomous differential-algebraic equations (DAEs). By using orthogonal changes of variables, the original DAE system is transformed into appropriate condensed forms, for which concepts such as Lyapunov exponents, Bohl exponents, exponential dichotomy and spectral intervals of various kinds can be analyzed via the resulting underlying ODE. Some essential differences between the spectral theory for ODEs and that for DAEs are pointed out. It is also discussed how numerical methods for computing the spectral intervals associated with Lyapunov and Sacker-Sell (exponential dichotomy) can be extended from those methods proposed for ODEs. Some numerical examples are presented to illustrate the theoretical results.

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“…Different sufficient conditions have been given in the framework of the tractability index [27,28,29,47] and the geometric index [32,37,40].…”

Section: Geometric Index Vs Kronecker Indexmentioning

confidence: 99%

Stability Loss in Quasilinear DAEs by Divergence of a Pencil Eigenvalue

Riaza¹

2010

SIAM J. Math. Anal.

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Abstract. The divergence through infinity of certain eigenvalues of a linearized differentialalgebraic equation (DAE) may result in a stability change along an equilibrium branch. This behavior cannot be exhibited by explicit ODEs, and its analysis in index one contexts has been so far unduly restricted to semiexplicit systems. By means of a geometric reduction framework we extend the characterization of this phenomenon to quasilinear DAEs, which comprise semiexplicit problems as a particular case. Our approach clarifies the nature of the singularities which are responsible for the stability change and also accommodates rank deficiencies in the leading system matrix. We show how to address this problem in a matrix pencil setting, an issue which leads to certain results of independent interest involving the geometric index of a quasilinear DAE and the Kronecker index of its linearization. The results are shown to be of interest in electrical circuit theory, since the differential-algebraic network models actually used in circuit simulation are not in semiexplicit but in quasilinear form.

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Criteria for the Trivial Solution of Differential Algebraic Equations with Small Nonlinearities to be Asymptotically Stable

Cited by 25 publications

References 14 publications

On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems

On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems

Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Stability Loss in Quasilinear DAEs by Divergence of a Pencil Eigenvalue

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