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

Linh, Vu Hoang; Mehrmann, Volker

doi:10.1007/s10884-009-9128-7

Cited by 39 publications

(80 citation statements)

References 60 publications

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“…The results here are easily modified to apply to the "adjoint" formulation of the discrete QR process Q n+1 R −T n = A −T n Q n . Finally, we note the recent work on stability spectrum for differential algebraic equations [16] and for noninvertible systems of linear difference equations [2] and the possibility of developing a quantitative perturbation theory based upon the ideas in this work.…”

Section: Discussionmentioning

confidence: 98%

On the Error in the Product QR Decomposition

Vleck¹

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We develop both a normwise and a componentwise error analysis for the QR factorization of long products of invertible matrices. We obtain global error bounds for both the orthogonal and upper triangular factors that depend on uniform bounds on the size of the local error, the local degree of nonnormality, and integral separation, a natural condition related to gaps between eigenvalues but for products of matrices. We illustrate our analytical results with numerical results that show the dependence on the degree of nonnormality and the strength of integral separation. 1. Introduction. In this paper we consider perturbation analysis for the QR decomposition of long products of matrices. In particular, we obtain uniform norm bounds and componentwise bounds on the orthogonal and upper triangular factors under the assumption of integral separation. Integral separation is a natural analogue for products of matrices to having gaps between eigenvalues of a matrix. Our approach is based upon ideas that are central to perturbation theory for Lyapunov exponents of linear nonautonomous differential equations. In particular, the results obtained here improve upon the results that can be obtained by combining the results in [8] and [11] by requiring less stringent assumptions and by obtaining sharper bounds. Here we avoid an intermediate step in which a perturbed triangular differential equation is obtained and work directly with perturbed triangular matrix products.We consider sources of perturbation error that include roundoff error, measurement error, and discretization error due, for example, to approximating the solution of a differential equation. The difficulty or conditioning of the problem is characterized by the degree of nonnormality and the strength of integral separation. The results we obtain allow for error bounds up to a certain size and structure in the perturbation error as compared with the conditioning of the problem.Our work here draws motivation from the work on QR and singular value decompositions of long matrix products (see, e.g., [22,20,18,14,15]) as well as the perturbation results for the QR factorization of a matrix, for example, [21,23,24,5,6]. We make use of the structural assumption of integral separation, which is central to the perturbation theory for Lyapunov exponents, stability spectra that play an analogous role to the real parts of eigenvalues for nonautonomous linear differential equations. An excellent reference that summarizes many results on Lyapunov exponents is the monograph by Adrianova [1]. In a series of papers [7,8,9,10,11], Dieci and the author have justified and developed an error analysis for approximation of Lyapunov exponents for nonautonomous linear differential equations via continuous QR factorizations of fundamental matrix solutions.

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Section: Discussionmentioning

confidence: 98%

On the Error in the Product QR Decomposition

Vleck¹

2010

SIAM J. Matrix Anal. & Appl.

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“…where A : R + 0 → R 2×2 is defined as in (19). Then the Bohl spectrum Bohl and the SackerSell spectrum SS of this system are given by Bohl …”

Section: Proposition 16mentioning

confidence: 99%

“…where A : R + 0 → R d×d is given as in (19), and f : Proof Since the Bohl spectrum is given by {−1}, both Lyapunov exponents must be −1. .…”

Section: Proposition 29 Consider the Nonlinear Differential Equatioṅmentioning

confidence: 99%

The Bohl Spectrum for Linear Nonautonomous Differential Equations

2016

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We develop the Bohl spectrum for nonautonomous linear differential equations on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker-Sell spectra. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker-Sell spectrum in general even for bounded systems. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable (which is not evident from the Sacker-Sell spectrum), but we show that in general this is not true. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker-Sell spectrum.

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“…Another important topic in which recently breakthroughs have been made is the stability analysis for DAEs, including the computation of Lyapunov exponents and Sacker-Sell spectra [32,33]. Here an open problem is the extension of these results to fully nonlinear systems as well as the development of improved computational methods for large scale problems.…”

Section: Recent Developments and Open Problemsmentioning

confidence: 99%

Control and Optimization with Differential-Algebraic Constraints

Biegler¹,

Campbell²,

Mehrmann³

2012

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Differential Algebraic Equations (DAEs) are mixed systems of differential and algebraic equations. It has been recognized for some time now that they have great potential both theoretically and in applications. DAEs form one of the most elegant and simple ways to model a physical system because they allow for the creation of separate models for subcomponents that can then be pasted together via a network. As a consequence, this concept is used in many modern CAD/modeling systems like SIMULINK, Scicos and DYMOLA, although most software packages cannot fully exploit the full potential of DAE models.But this nice feature of DAEs for modeling has also a disadvantage, since it shifts all of the difficulties of a system onto the analysis and the numerical methods. For this reason in recent years much effort has been spent to analyze general DAEs and to derive suitable numerical methods either for general DAEs, see e.g. [9,19,18,20,29,38] or for special DAEs arising in applications, see e.g. [6,12,28,37].This analytical and numerical work so far has been primarily driven by the simulation community, where the desire was to simulate the behavior of a complex system which could be electrical, mechanical, chemical, or all three. Due to the ever growing complexity of models which pose new challenges, the field is developing rather rapidly including now also hybrid [2,3,4,11,21,31,39] and delay systems [16,17,25,26,27].Once a system can be modeled and simulated, there arises the need to control the process or optimize its performance. The control of physical processes is an important task in many applications and over the last two decades there have been tremendous advances in the theory and applications of control in almost all disciplines of science and technologies. Note that this includes not only the obvious applications such as designing a more efficient process, but also determining what are the control mechanisms inherent in complex biological systems and fitting models to data. Recent Developments and Open ProblemsAs the two topics, simulation of DAEs and control/optimization, have evolved in recent years, there has been a growing awareness of interconnections which arise in a number of ways. One obvious way is the control of DAE modeled systems. Optimality conditions and a maximum principle for general DAEs have only recently been obtained [30] and the results are far from complete. But there are other more subtle connections in that the necessary conditions for an optimal control problem typically form a DAE and the solution of some control problems in the presence of constraints or invariants also involve DAEs. An open problem is in 1

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

Cited by 39 publications

References 60 publications

On the Error in the Product QR Decomposition

On the Error in the Product QR Decomposition

The Bohl Spectrum for Linear Nonautonomous Differential Equations

Control and Optimization with Differential-Algebraic Constraints

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