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Input-to-state stability in integral norms for linear infinite-dimensional systems

Sahiba Arora, Andrii Mironchenko

TL;DR

The paper advances the theory of integral-to-integral ISS for linear infinite-dimensional systems by introducing $L^p$-$L^q$ admissibility for unbounded input operators and establishing that integral-ISS is governed by exponential stability plus infinite-time admissibility, with maximal regularity providing a precise analytic framework for $p=q$. It demonstrates a nuanced divergence between finite-time and infinite-time admissibility in general, but shows equivalences in analytic settings and provides Lyapunov methods—coercive for bounded inputs and non-coercive via fractional extrapolation—for both bounded and unbounded control. Through diagonal and diffusion examples, the authors illustrate how Lyapunov constructions yield $L^p$-$L^p$ or $L^q$-$L^q$ ISS, and they extend the approach to fractional Sobolev spaces to handle unbounded input operators. Overall, the work connects maximal regularity, semigroup stability, admissibility, and Lyapunov theory to deliver a comprehensive framework for ISS in integral norms with broad applicability to boundary-control systems and diffusion processes.

Abstract

We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in $L^p$-spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded input operators. While input-to-state stability is typically characterized by exponential stability and finite-time admissibility, we show that this equivalence does not extend directly to integral norms. For analytic semigroups, we establish a precise characterization using maximal regularity theory. Additionally, we provide direct Lyapunov theorems and construct Lyapunov functions for $L^p$-$L^q$-ISS and demonstrate the results with examples, including diagonal systems and diffusion equations.

Input-to-state stability in integral norms for linear infinite-dimensional systems

TL;DR

The paper advances the theory of integral-to-integral ISS for linear infinite-dimensional systems by introducing - admissibility for unbounded input operators and establishing that integral-ISS is governed by exponential stability plus infinite-time admissibility, with maximal regularity providing a precise analytic framework for . It demonstrates a nuanced divergence between finite-time and infinite-time admissibility in general, but shows equivalences in analytic settings and provides Lyapunov methods—coercive for bounded inputs and non-coercive via fractional extrapolation—for both bounded and unbounded control. Through diagonal and diffusion examples, the authors illustrate how Lyapunov constructions yield - or - ISS, and they extend the approach to fractional Sobolev spaces to handle unbounded input operators. Overall, the work connects maximal regularity, semigroup stability, admissibility, and Lyapunov theory to deliver a comprehensive framework for ISS in integral norms with broad applicability to boundary-control systems and diffusion processes.

Abstract

We study integral-to-integral input-to-state stability for infinite-dimensional linear systems with inputs and trajectories in -spaces. We start by developing the corresponding admissibility theory for linear systems with unbounded input operators. While input-to-state stability is typically characterized by exponential stability and finite-time admissibility, we show that this equivalence does not extend directly to integral norms. For analytic semigroups, we establish a precise characterization using maximal regularity theory. Additionally, we provide direct Lyapunov theorems and construct Lyapunov functions for --ISS and demonstrate the results with examples, including diagonal systems and diffusion equations.
Paper Structure (9 sections, 16 theorems, 90 equations, 1 figure)

This paper contains 9 sections, 16 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear systems and their admissibility
  3. $L^p$-$L^q$-admissibility for linear systems
  4. $L^p$-$L^p$-admissibility for analytic semigroups
  5. Input-to-state stability in integral norms
  6. Lyapunov method
  7. Linear systems with bounded control operators
  8. Lyapunov functions for systems with unbounded control operators
  9. Construction of Lyapunov functions

Key Result

Proposition 2.4

The system $\Sigma(A,B)$ in Setting setting:system is $L^p$-$L^q$-admissible if and only if for each $t>0$ there exists $c(t)>0$ such that for all $u\in L^p([0,t], U)$.

Figures (1)

  • Figure 1: Relations between admissibility notions for fixed $p,q\in[1,\infty]$.

Theorems & Definitions (45)

  • Definition 2.2
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • Proposition 2.10
  • proof
  • ...and 35 more