@article{Hagiwara2024, 
author = {Tomomichi Hagiwara and Masaki Sugiyama},
title = {L    2        /        L    1   induced norm and Hankel norm analysis in sampled-data systems},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {2},
pages = {3035-3075},
keywords = {sampled-data systems, intersample behavior, induced norm, quasi Hankel norm, Hankel norm, H2 norm, impulse response},
url = {https://www.sciopen.com/article/10.3934/math.2024149},
doi = {10.3934/math.2024149},
abstract = {This paper is concerned with the        L    2        /        L    1   induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the    h-periodicity of the input-output relation of sampled-data systems is taken into account, where    h denotes the sampling period; past and future are separated by the instant    Θ  ∈  [  0  ,  h  ), and the norm of the operator describing the mapping from the past input in        L    1   to the future output in        L    2   is called the quasi        L    2        /        L    1   Hankel norm at    Θ. The        L    2        /        L    1   Hankel norm is defined as the supremum over    Θ  ∈  [  0  ,  h  ) of this norm, and if it is actually attained as the maximum, then a maximum-attaining    Θ is called a critical instant. This paper gives characterization for the        L    2        /        L    1   induced norm, the quasi        L    2        /        L    1   Hankel norm at    Θ and the        L    2        /        L    1   Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function    H  (  φ  ) on    [  0  ,  h  ) plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through    H  (  φ  ), even though    φ is a variable that is totally irrelevant to    Θ. The relevance of the induced/Hankel norm to the        H    2   norm of sampled-data systems is also discussed.}
}