The $ L_1 $-induced norm analysis for linear multivariable differential equations

Junghoon Kim; Jung Hoon Kim; Junghoon Kim; Jung Hoon Kim

doi:10.3934/math.20241629

AIMS Mathematics

2024, Volume 9, Issue 12: 34205-34223. doi: 10.3934/math.20241629

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The $ L_1 $-induced norm analysis for linear multivariable differential equations

Junghoon Kim ¹,
Jung Hoon Kim ^{1,2
,
,}

1.
Department of Electrical Engineering, Pohang University of Science and Technology, Pohang, 37673, Korea
2.
Institute for Convergence Research and Education in Advanced Technology, Yonsei University, Incheon 21983, Korea

Received: 20 September 2024 Revised: 26 November 2024 Accepted: 29 November 2024 Published: 04 December 2024
MSC : 39A06, 40A25, 45A05, 65L03, 93-08, 93C05, 93C55

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.
- linear multivariable differential equations,
- $ L_1 $-induced norm,
- $ l_1 $-induced norm,
- convergence analysis,
- operator approximations
Citation: Junghoon Kim, Jung Hoon Kim. The $ L_1 $-induced norm analysis for linear multivariable differential equations[J]. AIMS Mathematics, 2024, 9(12): 34205-34223. doi: 10.3934/math.20241629

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Abstract

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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