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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2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241629 |
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| author | Junghoon Kim Jung Hoon Kim |
| author_facet | Junghoon Kim Jung Hoon Kim |
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| description | 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. |
| format | Article |
| id | doaj-art-4d25c3c439bc49a9b020eec2d1d4aa62 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
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| spelling | doaj-art-4d25c3c439bc49a9b020eec2d1d4aa622025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912342053422310.3934/math.20241629The $ L_1 $-induced norm analysis for linear multivariable differential equationsJunghoon Kim0Jung Hoon Kim1Department of Electrical Engineering, Pohang University of Science and Technology, Pohang, 37673, KoreaDepartment of Electrical Engineering, Pohang University of Science and Technology, Pohang, 37673, KoreaIn 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 $-induced norm of difference equations was tackled in a parallel fashion. Finally, numerical studies were given to demonstrate the overall arguments.https://www.aimspress.com/article/doi/10.3934/math.20241629linear multivariable differential equations$ l_1 $-induced norm$ l_1 $-induced normconvergence analysisoperator approximations |
| spellingShingle | Junghoon Kim Jung Hoon Kim The $ L_1 $-induced norm analysis for linear multivariable differential equations AIMS Mathematics linear multivariable differential equations $ l_1 $-induced norm $ l_1 $-induced norm convergence analysis operator approximations |
| title | The $ L_1 $-induced norm analysis for linear multivariable differential equations |
| title_full | The $ L_1 $-induced norm analysis for linear multivariable differential equations |
| title_fullStr | The $ L_1 $-induced norm analysis for linear multivariable differential equations |
| title_full_unstemmed | The $ L_1 $-induced norm analysis for linear multivariable differential equations |
| title_short | The $ L_1 $-induced norm analysis for linear multivariable differential equations |
| title_sort | l 1 induced norm analysis for linear multivariable differential equations |
| topic | linear multivariable differential equations $ l_1 $-induced norm $ l_1 $-induced norm convergence analysis operator approximations |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241629 |
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