Vu Hoang Linh

Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions

Abstract This paper is concerned with the asymptotic stability of differential-algebraic equations with multiple delays and their numerical solutions. First, we give a sufficient condition for delay-independent stability. After characterizing the coefficient matrices that satisfy this stability condition, we propose some practical checkable criteria for asymptotic stability. Then we investigate the stability of numerical solutions obtained by θ-methods and BDF methods. Finally, solvability and stability of a class of weakly regular delay differential-algebraic equations are analyzed.

Implicit-system approach to the robust stability for a class of singularly perturbed linear systems

Abstract Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential-algebraic equations (DAEs) are studied. The asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero. The main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis, only. Our results can be extended to other singular perturbation problems for DAEs of more general form.

On the robust stability of implicit linear systems containing a small parameter in the leading term

This paper deals with the robust stability of implicit linear systems containing a small parameter in the leading term. Based on possible changes in the algebraic structure of the matrix pencils, a classification of such systems is given. The main attention is paid to the cases when the appearance of the small parameter causes some structure change in the matrix pencil. First, we give sufficient conditions providing the asymptotic stability of the parameterized system. Then, we give a formula for the complex stability radius and characterize its asymptotic behaviour as the parameter tends to zero. The structure-invariant cases are discussed, too. A conclusion concerning the parameter dependence of the robust stability is obtained.

STABILITY AND ROBUST STABILITY OF LINEAR TIME-INVARIANT DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS ∗

Necessary and sufficient conditions for exponential stability of linear time-invariant delay differential-algebraic equations are presented. The robustness of this property is studied when the equation is subjected to structured perturbations and a computable formula for the structured stability radius is derived. The results are illustrated by several examples.

Robust Stability of Differential-Algebraic Equations

This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated.

QR methods and error analysis for computing Lyapunov and Sacker---Sell spectral intervals for linear differential-algebraic equations

In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker–Sell spectral intervals for linear differential-algebraic equations. Furthermore, a perturbation and error analysis for these methods is presented. We investigate how errors in the data and in the numerical integration affect the accuracy of the approximate spectral intervals. Although we need to integrate numerically some differential-algebraic systems on usually very long time-intervals, under certain assumptions, it is shown that the error of the computed spectral intervals can be controlled by the local error of numerical integration and the error in solving the algebraic constraint. Some numerical examples are presented to illustrate the theoretical results.

Differential-algebraic equations

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Approximation of Spectral Intervals and Leading Directions for Differential-Algebraic Equation via Smooth Singular Value Decompositions

This paper is devoted to the numerical approximation of Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations (DAEs). The spectral analysis for DAEs is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to DAEs. Numerical methods based on smooth singular value decompositions are introduced for computing all or only some spectral intervals and their associated leading directions. The numerical algorithms as well as implementation issues are discussed in detail and numerical examples are presented to illustrate the theoretical results.

Efficient integration of strangeness-free non-stiff differential-algebraic equations by half-explicit methods

Numerical integration methods for nonlinear differential-algebraic equations (DAEs) in strangeness-free form are studied. In particular, half-explicit methods based on popular explicit methods like one-leg methods, linear multistep methods, and Runge-Kutta methods are proposed and analyzed. Compared with well-known implicit methods for DAEs, these half-explicit methods demonstrate their efficiency particularly for a special class of semi-linear matrix-valued DAEs which arise in the numerical computation of spectral intervals for DAEs. Numerical experiments illustrate the theoretical results.

Exponential Stability and Robust Stability for Linear Time-Varying Singular Systems of Second Order Difference Equations

In this paper, solvability, stability, and robust stability of linear time-varying singular systems of second order difference equations are studied. The leading coefficient is allowed to be singul...