A Remark on Matrix Equation on Small Time Scales

Adamec, Ladislav

doi:10.1080/10236190412331303468

Cited by 7 publications

(8 citation statements)

References 3 publications

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“…The following example is motivated by [8], Example 3. Consider the regressive (on T = Z) time varying homogeneous discrete linear dynamic system AX{t) = A{t)X{t), X{O) = I, (6,1) where.…”

Section: The Transition Matrix On Specific Time Scalesmentioning

confidence: 99%

“…Also, unlike [1] the development remains in the time scales setting throughout the paper. There is no need for carrying over to a corresponding ODE to obtain a principal fundamental matrix, and then bringing that solution back to the time scale domain.…”

Section: Introductionmentioning

confidence: 99%

“…x{t) = (/-(-M)<'~'''>/''xo). Adamec [1] has recently made an attempt to derive explicit formulae for the generalized matrix exponential for equation (1.1) when A{t) = A is a constant matrix and the transition matrix of (1.1) for time dependent A{t). The solution is derived by restricting a principal fundamental matrix Y{t, to) of an ordinary differential equation x' = H(t)x, t G.U..…”

Section: Introductionmentioning

confidence: 99%

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Transition matrix and generalized matrix exponential via the Peano-Baker series

DaCunha

2005

Journal of Difference Equations and Applications

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We give a closed form for the unique solution to the n X n regressive time varying linear dynamic system of the form x'^it) = A(t)x(t),x{to) = xg, via use of a newly developed generalized form of the PeanoBaker series. We develop a power series representation for the generalized time scale matrix exponential when the matrix A{t) = A is a constant matrix. We also introduce a finite series representation of the matrix exponential using the Laplace transform for time scales, as well as a theorem which allows us to write the matrix exponential as a series of (n -1) terms of scalar C^(T, R) functions multiplied by powers of the system matrix A.

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“…The following example is motivated by [8], Example 3. Consider the regressive (on T = Z) time varying homogeneous discrete linear dynamic system AX{t) = A{t)X{t), X{O) = I, (6,1) where.…”

Section: The Transition Matrix On Specific Time Scalesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transition matrix and generalized matrix exponential via the Peano-Baker series

DaCunha

2005

Journal of Difference Equations and Applications

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“…, t n } ⊆ [a, b] T , where a = t 0 < t 1 < · · · < t n = b. It is possible to prove ( [1] or [4]) the following lemma. T be a time scale, a, b ∈ T and a b. Then for every n ∈ N there exists a partition P n := {t 0 , t 1 , .…”

Section: Proof Since T < A(t) −1 On T the Spectral Radius Of µ(T)a(mentioning

confidence: 99%

“…Then, according to[1, Theorem 3.1], e A(·) (T , t 0 ) = lim Hence det e A(·) (T , t 0 ) = det lim Example 3.3. Let T be the middle third Cantor set, t 0 = 0, t = 1, and A ∈ Mat(d × d) be a constant matrix such that A < 1/3.…”

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confidence: 97%

A remark on Liouville's formula on small time scales

Adamec

2005

Journal of Mathematical Analysis and Applications

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Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

Hossain

Ghosh

2020

Numerical Meth Engineering

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SummaryMultiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12‐13): 741‐758) to adaptively construct a robust proper orthogonal decomposition‐based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed‐up in UQ is achieved.

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A Remark on Matrix Equation on Small Time Scales

Cited by 7 publications

References 3 publications

Transition matrix and generalized matrix exponential via the Peano-Baker series

Transition matrix and generalized matrix exponential via the Peano-Baker series

A remark on Liouville's formula on small time scales

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

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