The $ L_1 $-induced norm analysis for linear multivariable differential equations
In this paper, we consider the $ L_1 $-induced norm analysis for linear multivariable differential equations. Because such an analysis requires integrating the absolute value of the associated impulse response on the infinite-interval $ [0, \infty) $, this interval was divided into $ [0, H) $ and $...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241629 |
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| Summary: | In this paper, we consider the $ L_1 $-induced norm analysis for linear multivariable differential equations. Because such an analysis requires integrating the absolute value of the associated impulse response on the infinite-interval $ [0, \infty) $, this interval was divided into $ [0, H) $ and $ [H, \infty) $, with the truncation parameter $ H $. The former was divided into $ M $ subintervals with an equal width, and the kernel function of the relevant input\slash output operator on each subinterval was approximated by a $ p $th order polynomial with $ p = 0, 1, 2, 3 $. This derived to an upper bound and a lower bound on the $ L_1 $-induced norm for $ [0, H) $, with the convergence rate of $ 1/M^{p+1} $. An upper bound on the $ L_1 $-induced norm for $ [H, \infty) $ was also derived, with an exponential order of $ H $. Combining these bounds led to an upper bound and a lower bound on the original $ L_1 $-induced norm on $ [0, \infty) $, within the order of $ 1/M^{p+1} $. Furthermore, the $ l_1 $-induced norm of difference equations was tackled in a parallel fashion. Finally, numerical studies were given to demonstrate the overall arguments. |
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| ISSN: | 2473-6988 |