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Two energy methods for distributed port-Hamiltonian systems and their application to stability analysis

Marco Roschkowski, Hannes Gernandt

TL;DR

Addressing stability of distributed-parameter port-Hamiltonian systems on one-dimensional domains, the paper develops two local energy methods that relate exponential stability to boundary energy content over a time horizon. The authors introduce energy functionals that either integrate energy along space-time slabs or along time-windows, derive monotonicity properties, and prove an equivalence: exponential stability iff a boundary-energy inequality holds at an end-point over a suitable time window. They apply the results to networks of vibrating strings, successfully verifying exponential stability in two- and four-string configurations where existing sufficient conditions fail. They also discuss short-time behavior, showing no universal short-time decay and connecting the analysis to hypocoercivity concepts. The work provides practical criteria for stability analysis of pH networks and paves the way for analyzing more complex networks of distributed parameter systems.

Abstract

We develop two local energy methods for distributed parameter port-Hamiltonian (pH) systems on one-dimensional spatial domains. The methods are applied to derive a characterization of exponential stability directly in terms of the energy passing through the boundary over a given time horizon. The resulting condition is verified for a network of vibrating strings where existing sufficient conditions cannot be applied. Moreover, we use a local energy method to study the short-time behavior of pH systems with boundary damping which was recently studied in the context of hypocoercivity.

Two energy methods for distributed port-Hamiltonian systems and their application to stability analysis

TL;DR

Addressing stability of distributed-parameter port-Hamiltonian systems on one-dimensional domains, the paper develops two local energy methods that relate exponential stability to boundary energy content over a time horizon. The authors introduce energy functionals that either integrate energy along space-time slabs or along time-windows, derive monotonicity properties, and prove an equivalence: exponential stability iff a boundary-energy inequality holds at an end-point over a suitable time window. They apply the results to networks of vibrating strings, successfully verifying exponential stability in two- and four-string configurations where existing sufficient conditions fail. They also discuss short-time behavior, showing no universal short-time decay and connecting the analysis to hypocoercivity concepts. The work provides practical criteria for stability analysis of pH networks and paves the way for analyzing more complex networks of distributed parameter systems.

Abstract

We develop two local energy methods for distributed parameter port-Hamiltonian (pH) systems on one-dimensional spatial domains. The methods are applied to derive a characterization of exponential stability directly in terms of the energy passing through the boundary over a given time horizon. The resulting condition is verified for a network of vibrating strings where existing sufficient conditions cannot be applied. Moreover, we use a local energy method to study the short-time behavior of pH systems with boundary damping which was recently studied in the context of hypocoercivity.

Paper Structure

This paper contains 8 sections, 6 theorems, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Energy methods for port-Hamiltonian systems
  4. Applications of energy methods to long- and short-time behavior
  5. Verification of exponential stability for vibrating string networks
  6. Two Interconnected strings
  7. Four interconnected strings
  8. Conclusion

Key Result

Theorem 1

Consider a pH system pH that satisfies Assumption assumption with $(\bm{T}(t))_{t \ge 0}$ being the semigroup generated by $\bm{A}$. If there exists $k > 0$ such that one of the following conditions hold for all $x\in D(\bm{A})$ then $(\bm{T}(t))_{t\ge 0}$ is exponentially stable.

Figures (4)

  • Figure 1: Visual sketch of the function $\bm{F}$. The function $\tilde{\bm{F}}$ can formally be obtained by flipping the sketch horizontally.
  • Figure 2: Visual sketch of the functions $\bm{G}$ and $\tilde{\bm{G}}=\tilde{\bm{G}}_1+\tilde{\bm{G}}_2$.
  • Figure 3: A simple example where Theorem \ref{['stability basic']} can be applied, but Theorem \ref{['thm: birgit thm']} not.
  • Figure 4: A more complicated string network

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 2
  • proof
  • Proposition 4
  • ...and 1 more