On well‐definability of the L ∞ / L 2 Hankel operator and detection of all the critical instants in sampled‐data systems

Abstract

Because sampled‐data systems have h‐periodic nature with the sampling period h, an arbitrary Θ∈[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 norm called the quasi 𝐿∞/𝐿2 Hankel norm at Θ is used to define the 𝐿∞/𝐿2 Hankel norm as the supremum of their values over Θ∈[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.