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Infinite-dimensional port-Hamiltonian systems with a stationary interface

Alexander Kilian, Bernhard Maschke, Andrii Mironchenko, Fabian Wirth

TL;DR

The paper develops a mathematically rigorous, infinite-dimensional port-Hamiltonian formulation for two one-dimensional conservation-law systems on complementary intervals coupled by a stationary interface. It introduces color functions, interface ports, and a skew-symmetric interconnection operator to obtain a balance law and establishes conditions under which the coupled system generates a contraction semigroup, with additional criteria yielding exponential stability. The results are demonstrated on a coupled acoustic waveguide example and extended to multiple interfaces, with discussions of moving-interface extensions. Overall, the framework provides practical, checkable passivity conditions for well-posedness and stability of interfaced port-Hamiltonian systems, and lays groundwork for moving-interface analysis and broader interconnections in distributed-parameter settings.

Abstract

We consider two systems of two conservation laws that are defined on complementary, one-dimensional spatial intervals and coupled by an interface as a single port-Hamiltonian system. In case of a fixed interface position, we characterize the boundary and interface conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. Furthermore, we present sufficient conditions for the exponential stability of the generated $C_0$-semigroup. The results are illustrated by the example of two acoustic waveguides coupled by a membrane interface.

Infinite-dimensional port-Hamiltonian systems with a stationary interface

TL;DR

The paper develops a mathematically rigorous, infinite-dimensional port-Hamiltonian formulation for two one-dimensional conservation-law systems on complementary intervals coupled by a stationary interface. It introduces color functions, interface ports, and a skew-symmetric interconnection operator to obtain a balance law and establishes conditions under which the coupled system generates a contraction semigroup, with additional criteria yielding exponential stability. The results are demonstrated on a coupled acoustic waveguide example and extended to multiple interfaces, with discussions of moving-interface extensions. Overall, the framework provides practical, checkable passivity conditions for well-posedness and stability of interfaced port-Hamiltonian systems, and lays groundwork for moving-interface analysis and broader interconnections in distributed-parameter settings.

Abstract

We consider two systems of two conservation laws that are defined on complementary, one-dimensional spatial intervals and coupled by an interface as a single port-Hamiltonian system. In case of a fixed interface position, we characterize the boundary and interface conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. Furthermore, we present sufficient conditions for the exponential stability of the generated -semigroup. The results are illustrated by the example of two acoustic waveguides coupled by a membrane interface.
Paper Structure (12 sections, 14 theorems, 105 equations)

This paper contains 12 sections, 14 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Port-Hamiltonian Systems with Interface
  3. Model Description
  4. Derivation of the Balance Equations
  5. Port-Hamiltonian Formulation
  6. Generation of a Contraction Semigroup
  7. Boundary and Interface Conditions
  8. A Reformulation as a Boundary Port-Hamiltonian System
  9. Exponential Stability
  10. Example: Coupled Acoustic Waveguides
  11. Port-Hamiltonian Systems with Multiple Interfaces
  12. Conclusions and Extensions Towards Moving Interfaces

Key Result

Proposition 1

Let $x$ be defined as in aggregate state x, where $x^{\pm}$ are weak solutions of the systems of conservation laws Systems of Conservation Laws, satisfying the continuity condition Interface Variable fI. Then equation (Conservation Law of x1) holds in a distributional sense for all $t >0$, that is,

Theorems & Definitions (25)

  • Proposition 1
  • proof
  • Proposition 2
  • remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • remark 2: Dirac Structure of \ref{['Simplified PH-System']}
  • Lemma 5
  • ...and 15 more