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Estimation of the input-to-state stability gain functions from finite-dimensional approximations

Birgit Hillebrecht, Benjamin Unger

Abstract

Since the concept of input-to-state stability (ISS) was introduced, it has been extensively investigated for finite-dimensional control systems and has recently received attention for infinite-dimensional systems. While numerical techniques provide a bridge between these two worlds, a rigorous connection between the ISS of an infinite-dimensional system with an unbounded control operator and the properties of its finite-dimensional approximations has not yet been established. In this manuscript, we make a first step towards closing this gap by investigating numerical approximations of linear (boundary) control systems using semigroup theory. Specifically, we focus on linear boundary control systems where the autonomous evolution is governed by an analytic semigroup. For these systems, we show that ISS gains can be computed from approximations. We illustrate the applicability of these findings using a one-dimensional heat equation with Dirichlet boundary control for which reference ISS gains are known.

Estimation of the input-to-state stability gain functions from finite-dimensional approximations

Abstract

Since the concept of input-to-state stability (ISS) was introduced, it has been extensively investigated for finite-dimensional control systems and has recently received attention for infinite-dimensional systems. While numerical techniques provide a bridge between these two worlds, a rigorous connection between the ISS of an infinite-dimensional system with an unbounded control operator and the properties of its finite-dimensional approximations has not yet been established. In this manuscript, we make a first step towards closing this gap by investigating numerical approximations of linear (boundary) control systems using semigroup theory. Specifically, we focus on linear boundary control systems where the autonomous evolution is governed by an analytic semigroup. For these systems, we show that ISS gains can be computed from approximations. We illustrate the applicability of these findings using a one-dimensional heat equation with Dirichlet boundary control for which reference ISS gains are known.
Paper Structure (12 sections, 10 theorems, 82 equations, 5 figures)

This paper contains 12 sections, 10 theorems, 82 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and problem statements
  3. Sectorial operators, analytic semigroups, and fractional powers
  4. Control systems and input-to-state stability
  5. Approximation of strongly continuous semigroups of operators
  6. Problem statements
  7. Approximation of the ISS gain $\gamma$ for control systems
  8. Approximation of ISS gains for boundary control systems
  9. Motivational example
  10. Convergence of the boundary operator
  11. Numerical example: Dirichlet controlled heat equation
  12. Discussion

Key Result

Lemma 2.6

Consider a sectorial operator $-\mathcal{A}$. Then for $\alpha < \beta < \gamma$, there exists an $L > 0$ such that

Figures (5)

  • Figure 1: Illustration of the spectrum $\sigma(-\mathcal{A})$ and the sector $S_{\varsigma}$ from the definition of a sectorial operator $-\mathcal{A}$ and the integration path of the definition of fractional powers.
  • Figure 2: Modelling of finite difference discretization. The red areas depict the intervals fully determined by the boundary conditions.
  • Figure 3: Systems under consideration.
  • Figure 4: Fattorini trick applied to systems (C) and (D) from \ref{['fig:bcsystems']}.
  • Figure 5: Values for the growth bound, the resolvent bound, and the norm of the boundary operator in the fractional power space for various equidistant discretizations of $[0, 1]$ into $n$ intervals.

Theorems & Definitions (34)

  • Definition 2.1: Haa03
  • Definition 2.2: Haa03
  • Definition 2.3: EngN00
  • Definition 2.4: EngN00
  • Definition 2.5
  • Lemma 2.6: EngN00
  • Definition 2.7: Lun12
  • Definition 2.8: Haa03
  • Definition 2.9: EngN00
  • Definition 2.10: Sch20
  • ...and 24 more