Lineare Algebra

Niemeyer, Horst; Wermuth, Edgar

doi:10.1007/978-3-322-83034-0

Cited by 20 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [3,[17][18][19][20][21][22] the reader finds dynamical problems of interest with respect to the present paper. In the References [23][24][25][26] there is some material on Linear Algebra useful in the context of the paper. The references [27] and [28] are on functional analytical methods used in the author's work.…”

Section: Comments On the Referencesmentioning

confidence: 99%

Two-Sided Bounds on the Displacement y(t) and the Velocity ý(t) of the Vibration Problem Mÿ+Bý + Ky = 0,y(t0)= y0,ý(t0) ý0= with Application of the Differential Calculus of Norms

Kohaupt¹

2011

TOAMJ

View full text Add to dashboard Cite

If the vibration problem M y + B y + Ky = 0,y(t 0 ) = y 0 , y(t 0 ) = y 0 , is cast into state-space form x = Ax,x(t 0 ) = x 0 , so far only two-sided bounds on x(t) could be derived, but not on the quantities y(t) and y(t) . By means of new methods, this gap is now filled by deriving two-sided bounds on y(t) and y(t) ; they have the same shape as those for x(t) . The best constants in the upper bounds are computed by the differential calculus of norms developed by the author in earlier work. As opposed to this, the lower bounds cannot be determined in the same way since y(t) 2 and y(t) 2 have kinks at their local minima (like | t | 1/2 at t = 0 ). The best lower bounds are therefore determined through their local minima. The obtained results cannot be obtained by the methods used so far.

show abstract

Section: Comments On the Referencesmentioning

confidence: 99%

Two-Sided Bounds on the Displacement y(t) and the Velocity ý(t) of the Vibration Problem Mÿ+Bý + Ky = 0,y(t0)= y0,ý(t0) ý0= with Application of the Differential Calculus of Norms

Kohaupt¹

2011

TOAMJ

View full text Add to dashboard Cite

show abstract

“…Refs. [1][2][3][4]6,7,[9][10][11][12]24,[27][28][29][30][31][32][34][35][36] are given even though they are not directly used in this paper in order to provide the reader with some additional material useful in the discussed subject.…”

Section: Introductionmentioning

confidence: 99%

Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

Kohaupt

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

The new idea is to study the stability behavior of the solution x = x(t) of the initial value problemẋ = Ax, t t 0 , x(t 0 ) = x 0 , with A ∈ C n×n , in a weighted (semi-) norm · R where R is taken as an appropriate solution of the matrix eigenvalue problem RA + A * R = R, rather than as the solution of the algebraic Lyapunov matrix equation RA + A * R = −S with given positive (semi-) definite matrix S. Substantially better results are obtained by the new method. For example, if A is diagonalizable and all eigenvalues, which is much more than the old result, which only states that lim t→∞ x(t) = 0. Especially, the semi-norms · R i have a decoupling and filter effect on x(t). Further, new two-sided bounds (depending on x 0 ) for the asymptotic behavior can be derived. The best constants in the bounds are obtained by the differential calculus of norms. Applications are made to free linear dynamical systems, and computations underpin the theoretical findings. : Matrix eigenvalue problem VA + A * V = V ; Algebraic Lyapunov equation RA + A * R = −S; Linear dynamical system; Decoupling and filter effect of weighted semi-norms; Stability; Two-sided bounds depending on the initial conditions; Differential calculus of norms

show abstract

Literatur

1999

Persistenz Und Reichweite Von Umweltchemikalien

View full text Add to dashboard Cite

Lineare Algebra

Cited by 20 publications

References 0 publications

Two-Sided Bounds on the Displacement y(t) and the Velocity ý(t) of the Vibration Problem Mÿ+Bý + Ky = 0,y(t0)= y0,ý(t0) ý0= with Application of the Differential Calculus of Norms

Two-Sided Bounds on the Displacement y(t) and the Velocity ý(t) of the Vibration Problem Mÿ+Bý + Ky = 0,y(t0)= y0,ý(t0) ý0= with Application of the Differential Calculus of Norms

Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

Literatur

Contact Info

Product

Resources

About