@article{Kim2024, 
author = {Junghoon Kim and Jung Hoon Kim},
title = {The  L1-induced norm analysis for linear multivariable differential equations},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {12},
pages = {34205-34223},
keywords = {convergence analysis, linear multivariable differential equations, L1-induced norm, l1-induced norm, operator approximations},
url = {https://www.sciopen.com/article/10.3934/math.20241629},
doi = {10.3934/math.20241629},
abstract = {In this paper, we consider the  L1-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,∞), this interval was divided into  [0,H) and  [H,∞), 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  pth order polynomial with  p=0,1,2,3. This derived to an upper bound and a lower bound on the  L1-induced norm for  [0,H), with the convergence rate of  1/Mp+1. An upper bound on the  L1-induced norm for  [H,∞) was also derived, with an exponential order of  H. Combining these bounds led to an upper bound and a lower bound on the original  L1-induced norm on  [0,∞), within the order of  1/Mp+1. Furthermore, the  l1-induced norm of difference equations was tackled in a parallel fashion. Finally, numerical studies were given to demonstrate the overall arguments.}
}