On well-definability of the L∞/L₂ Hankel operator and detection of all the critical instants in sampled-data systems

Because sampled-data systems have h-periodic nature with the sampling period h, an arbitraryΘ ∈ [0, h) is taken and the quasi L∞∕L₂ Hankel operator at Θ is defined as the mapping from L₂(−∞, Θ) to L∞[Θ, ∞). Its normcalled the quasi L∞∕L₂ Hankel normat Θ is used to define the L∞∕L₂ Hankel norm as the supremum of their values overΘ ∈ [0, h). If the supremum is actually attained as the maximum, then a maximum-attaining Θ is called a critical instant and the L∞∕L₂ Hankel operator is said to be well-definable. An earlier study establishes a computation method of the L∞∕L₂ Hankel norm, which is called a sophisticated method if our interest lies only in its computation. However, the feature of the method that it is free from considering the quasi L∞∕L₂ Hankel norm for any Θ ∈ [0, h) prevents the earlier study to give any arguments as to whether the obtained L∞∕L₂ Hankel norm is actually attained as the maximum, as well as detecting all the critical instants when the L∞∕L₂ Hankel operator is well-definable. This paper establishes further arguments to tackle these relevant questions and provides numerical examples to validate the arguments in different aspects of authors' theoretical interests