Differential calculus for p-norms of complex-valued vector functions with applications

Kohaupt, L.

doi:10.1016/s0377-0427(01)00594-5

Cited by 13 publications

(11 citation statements)

References 14 publications

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“…, n. As a consequence, the semi-norms · R i have a decoupling and filter effect on the solution of the above IVP. Based hereupon, new two-sided bounds (depending on the initial condition x(t 0 ) = x 0 ) for the asymptotic behavior of the solution x = x(t) can be derived which considerably improve earlier estimates in [16]. Numerical computations confirm the theoretical results.…”

Section: Introductionsupporting

confidence: 52%

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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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Section: Introductionsupporting

confidence: 52%

“…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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“…Note, that the parameters ρ A and M A with respect to the spectral bound are not unique, especially any Bernstein ellipse can be chosen since the function is entire. Here, we chose ρ A with respect to the decay of the Chebyshev coefficients |c k | given by (16) but for the sake of simplicity the derivation is omitted and for the appropriate examples ρ A is given in Table 3. M A is determined by the strategy mentioned above, i.e., by a combination of fminsearch and Global Search.…”

Section: Overview and Numerical Resultsmentioning

confidence: 99%

“…3.1 we summarize results for linear time-invariant systems [18,19]. Two-sided bounds on the solution with the differential calculus of norms, e.g., in [15][16][17], are shown. In Sect.…”

Section: Introductionmentioning

confidence: 99%

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Benner

Denißen

Kohaupt

2016

J. Appl. Math. Comput.

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Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet's Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet's Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.

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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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Differential calculus for p-norms of complex-valued vector functions with applications

Cited by 13 publications

References 14 publications

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

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

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

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

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