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Solvability of time-varying infinite-dimensional linear port-Hamiltonian systems

Mikael Kurula

Abstract

Thirty years after the introduction of port-Hamiltonian systems, interest in this system class still remains high among systems and control researchers. Very recently, Jacob and Laasri obtained strong results on the solvability and well-posedness of time-varying linear port-Hamil\-to\-nian systems with boundary control and boundary observation. In this paper, we complement their results by discussing the solvability of linear, infinite-dimensional time-varying port-Hamiltonian systems not necessarily of boundary control type. The theory is illustrated on a system with a delay component in the state dynamics.

Solvability of time-varying infinite-dimensional linear port-Hamiltonian systems

Abstract

Thirty years after the introduction of port-Hamiltonian systems, interest in this system class still remains high among systems and control researchers. Very recently, Jacob and Laasri obtained strong results on the solvability and well-posedness of time-varying linear port-Hamil\-to\-nian systems with boundary control and boundary observation. In this paper, we complement their results by discussing the solvability of linear, infinite-dimensional time-varying port-Hamiltonian systems not necessarily of boundary control type. The theory is illustrated on a system with a delay component in the state dynamics.
Paper Structure (8 sections, 8 theorems, 53 equations)

This paper contains 8 sections, 8 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Dirac nodes and system nodes
  3. Time-varying port-Hamiltonian systems
  4. Time-varying well-posed systems
  5. Existence of classical solutions for impedance representations
  6. The example
  7. Conclusions
  8. Acknowledgment

Key Result

Theorem 2.5

Let $\left[\right]$ be a system node, let $e\in H^1_{loc}({{\mathbb R}_{+}};\mathcal{E})$ and $\left[\right]\in\mathrm{dom}\left(\left[\right]\right)$. Then there exist $x\in C^1({{\mathbb R}_{+}};\mathcal{X})$ and $f\in H^1_{loc}({{\mathbb R}_{+}};\mathcal{F})$, such that eq:LTI holds.

Theorems & Definitions (20)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 10 more