Differential calculus for some p-norms of the fundamental matrix with applications

Kohaupt, L.

doi:10.1016/s0377-0427(00)00559-8

Cited by 19 publications

(7 citation statements)

References 2 publications

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“…Some other upper bounds have been developed that could be utilized [7,11,20], but these are usually more computationally intensive than the simple ones used here. …”

Section: Definition 22mentioning

confidence: 99%

Computing the maximum transient energy growth

Whidborne

Amar

2011

Bit Numer Math

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“…Some other upper bounds have been developed that could be utilized [7,11,20], but these are usually more computationally intensive than the simple ones used here. …”

Section: Definition 22mentioning

confidence: 99%

Computing the maximum transient energy growth

Whidborne

Amar

2011

Bit Numer Math

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“…The References [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16] contain the most important contributions to the author's current research area. In [3,[17][18][19][20][21][22] the reader finds dynamical problems of interest with respect to the present paper.…”

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

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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.

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“…with the mass, damping, and stiffness matrices M, B, and K, as the case may be, and the displacement vector y as in [15]. In state-space description, this problem takes the forṁ…”

Section: The Multi-mass Vibration Problemmentioning

confidence: 99%

“…We compare old upper bounds with the new upper bounds depending on x 0 , and also give new lower bounds on x(t). The optimal constants in the bounds are computed by the differential calculus of norms developed by the author in earlier work, see [14][15][16][17][18][19][20][21]. In Section 6, we draw some conclusions.…”

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

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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

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Differential calculus for some p-norms of the fundamental matrix with applications

Cited by 19 publications

References 2 publications

Computing the maximum transient energy growth

Computing the maximum transient energy growth

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

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