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Boundary control systems on a one-dimension spatial domain

Bouchra Elghazi, Birgit Jacob, Hans Zwart

TL;DR

The paper tackles well-posed boundary control and observation systems on a one-dimensional spatial domain by deriving a simple, verifiable condition for well-posedness within a port-Hamiltonian framework. It shows that well-posedness under full control and observation entails exact controllability and exact observability, and it develops robustness results under bounded perturbations and admissible feedback. The theory is illustrated with Euler-Bernoulli beam models, including viscous damping and elastic-support scenarios, demonstrating practical applicability to flexible structures. A key perspective is extending the framework to more general beam models, such as the Rayleigh beam.

Abstract

The aim of this paper is to investigate the well-posedness of a class of boundary control and observation systems on a one dimensional spatial domain. We derive a necessary and sufficient condition characterizing the well-posedness of these systems. Furthermore, we show that the well-posedness and full control and observation implies exact controllability and exact observability. The theoretical results are illustrated using Euler-Bernoulli beam models.

Boundary control systems on a one-dimension spatial domain

TL;DR

The paper tackles well-posed boundary control and observation systems on a one-dimensional spatial domain by deriving a simple, verifiable condition for well-posedness within a port-Hamiltonian framework. It shows that well-posedness under full control and observation entails exact controllability and exact observability, and it develops robustness results under bounded perturbations and admissible feedback. The theory is illustrated with Euler-Bernoulli beam models, including viscous damping and elastic-support scenarios, demonstrating practical applicability to flexible structures. A key perspective is extending the framework to more general beam models, such as the Rayleigh beam.

Abstract

The aim of this paper is to investigate the well-posedness of a class of boundary control and observation systems on a one dimensional spatial domain. We derive a necessary and sufficient condition characterizing the well-posedness of these systems. Furthermore, we show that the well-posedness and full control and observation implies exact controllability and exact observability. The theoretical results are illustrated using Euler-Bernoulli beam models.
Paper Structure (8 sections, 17 theorems, 113 equations)

This paper contains 8 sections, 17 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Boundary control and observation systems
  3. Analysis of well-posedness
  4. Exact controllability and exact observability
  5. Examples
  6. Euler-Bernoulli beam with viscous air damping
  7. Beam with distributed elastic support
  8. Conclusion and perspectives

Key Result

Theorem 1

CurtZwa:20 Consider the boundary control and observation system BCS and the abstract differential equation BCS_linear with the output output_y_v. Assume that $u \in C^2([0,T]; U)$. Then, if $v_0 = x_0 - Bu(0) \in D(A_0)$, the classical solutions of BCS and BCS_linear--output_y_v are related by Furthermore, the classical solution of BCS is unique.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Theorem 1
  • Definition 2.3
  • Remark 2.1
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1
  • Corollary 2
  • Proof 1
  • ...and 27 more