Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Admissible operators for sun-dual semigroups

Sahiba Arora, Felix L. Schwenninger

TL;DR

The paper extends Weiss's duality for admissible operators to the sun-dual framework, addressing cases where the dual semigroup is not strongly continuous and thereby enabling admissibility analysis in nonreflexive spaces such as $C(K)$ and $L^1$. It establishes core duality results: $\mathrm{C}$-admissibility of control operators corresponds to $L^1$-admissibility of their sun-duals, and dual results for observation operators are developed with precise conditions for $p$ and $q$; these are complemented by a detailed example on a right-shift semigroup that concretely characterizes limit-case ($L^1$) admissibility via measures of bounded variation. The article further discusses generalizations beyond the sun-dual space, including moon-dual spaces for sectorial generators, and outlines implications for controllability and observability within this framework, pointing to rich directions for future research in nonreflexive and positive-system settings.

Abstract

We extend classical duality results by Weiss on admissible operators to settings where the dual semigroup lacks strong continuity. This is possible using the sun-dual framework, which is not immediate from the duality of the input and output maps. This extension enables the testing of admissibility for a broader range of examples, in particular for state space of continuous functions or $L^1$.

Admissible operators for sun-dual semigroups

TL;DR

The paper extends Weiss's duality for admissible operators to the sun-dual framework, addressing cases where the dual semigroup is not strongly continuous and thereby enabling admissibility analysis in nonreflexive spaces such as and . It establishes core duality results: -admissibility of control operators corresponds to -admissibility of their sun-duals, and dual results for observation operators are developed with precise conditions for and ; these are complemented by a detailed example on a right-shift semigroup that concretely characterizes limit-case () admissibility via measures of bounded variation. The article further discusses generalizations beyond the sun-dual space, including moon-dual spaces for sectorial generators, and outlines implications for controllability and observability within this framework, pointing to rich directions for future research in nonreflexive and positive-system settings.

Abstract

We extend classical duality results by Weiss on admissible operators to settings where the dual semigroup lacks strong continuity. This is possible using the sun-dual framework, which is not immediate from the duality of the input and output maps. This extension enables the testing of admissibility for a broader range of examples, in particular for state space of continuous functions or .
Paper Structure (8 sections, 10 theorems, 64 equations, 2 figures)

This paper contains 8 sections, 10 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Characterisation of $\mathrm C$-admissibility of control operators
  3. Duality results for control operators
  4. Duality results for observation operators
  5. An Example
  6. Concluding remarks: Generalizations and further directions
  7. Beyond the sun dual space
  8. Controllability and observability

Key Result

Proposition 2.1

Let $X$ and $U$ be Banach spaces, $(T(t))_{t\ge 0}$ be a $C_0$-semigroup on $X$ with generator $A$, let $\lambda \in \rho(A)$, and let $\tau>0$. For the control operator $B\in\mathcal{L}(U,X_{-1})$, the following are equivalent. Moreover, setting $F(\mathord{\,\cdot\,}):=T(\mathord{\,\cdot\,})R(\lambda,A_{-1})B$, we have that and for all $x' \in X'$ with $\left\lVert x' \right\rVert\le 1$.

Figures (2)

  • Figure 1: Duality between control operators $B\in \mathcal{L}(U, X_{-1})$ and observation operators $B'\in \mathcal{L}\left((X^{\odot})_{1},U'\right)$
  • Figure 2: Duality for observation operators $C\in \mathcal{L}(X_1, Y)$ with $\mathop{\mathrm{Rg}}\nolimits C' \subseteq \left(X^{\odot}\right)_{-1}$ and control operators $B'\in \mathcal{L}\left(Y',\left(X^{\odot}\right)_{-1}\right)$.

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm:sun-dual-continuous']}
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm:weiss-sun-dual-control']}
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • ...and 11 more