Dichotomies and stability theory

Coppel, W. A.

doi:10.1007/bfb0070202

Cited by 357 publications

(630 citation statements)

References 3 publications

Supporting

Mentioning

608

Contrasting

Unclassified

Order By: Relevance

“…The following examples show that, if one of the conditions (17), (19) fails to hold, then the DAE and its adjoint may be Lyapunov-regular, but the Perron identity (20) does not hold and also the converse implication may not hold. It is also possible that the Lyapunov-regularity of a DAE system does not imply the Lyapunov-regularity of its adjoint and vice versa.…”

Section: Remark 21mentioning

confidence: 98%

“…However, the validity of (17) is invariant under this transformation. Using the relation between the coefficients of (9) and those of (13), which follows by the proof of Theorem 12, one may easily reformulate the conditions (17), (19) in term of the original data, i.e., the coefficients of (9). Of course, the derivative of the matrix function Q appearing in Theorem 12 will be involved in the reformulated conditions.…”

Section: Remark 26mentioning

confidence: 99%

“…In the nonlinear case, by linearizing along a particular solution, Lyapunov exponents also give information about the convergence or divergence rates of nearby solutions. The spectral theory for ODEs was further developed throughout the 20th century, and concepts such as Bohl exponents, exponential dichotomy (also well-known as Sacker-Sell) spectra were introduced, see [1,19,20,70]. Unlike the development of the analytic theory, the development of numerical methods to compute Lyapunov exponents and also other spectral intervals has only recently been studied.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Linh

Mehrmann

2009

J Dyn Diff Equat

View full text Add to dashboard Cite

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.

show abstract

Section: Remark 21mentioning

confidence: 98%

Section: Remark 26mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Linh

Mehrmann

2009

J Dyn Diff Equat

View full text Add to dashboard Cite

show abstract

“…The alternative definition, as given in [3] can be formulated as follows: there exist projection-operator P and numbers K; > 0, such that for 8a; b 2 S k˝a 0 .A/P ˝b 0 .A/…”

Section: Introductionmentioning

confidence: 99%

A new criterion for the roughness of exponential dichotomy on R

Feketa¹,

Леонтьєв²

2015

Miskolc Math Notes

View full text Add to dashboard Cite

kB.t/k D 0. However, it is also known that corresponding statement for exponential dichotomy on R fails. In this work we show that under additional hypothesis of commutativity of A.s/ and B.t/ for all s; t 2 R it can indeed be shown that dichotomy on R is unaffected by "small" perturbations (while "small" now has meaning of "being in L 1 .R/ class"). We also provide some examples and particular corollaries of theory to show that hypotheses are natural and apply to wide range of problems.

show abstract

“…In the present paper one of the important properties of the ordinary dichotomy for linear impulsive differential equations is studied, namely that it is not destroyed under small perturbations of the coefficient matrix. We shall note that analogous questions about ordinary differential equations were considered in [7], [2], [1], [9] (6) where The operator Q is bounded The supplementary projector I -P = (I -P)(7 + QP) has a range ~(7 -P) = Z.…”

mentioning

confidence: 99%