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Extension of Controllability Score to Infinite-Dimensional Systems

Yuito Nakabe, Kazuhiro Sato

TL;DR

The paper tackles the problem of defining centrality in infinite-dimensional dynamical systems by extending two controllability scores, the Volumetric Controllability Score (VCS) and the Average Energy Controllability Score (AECS), through optimization over a diagonal input structure with weights summing to one. It proves the existence of optimal solutions under weak assumptions and establishes uniqueness under additional structural conditions, using the controllability Gramian $W(p)$ in the infinite-dimensional setting and analyzing eigenvalue-based objective forms. The authors provide a detailed special-case analysis for self-adjoint generators and demonstrate the approach with heat-equation-based numerical experiments, showing how VCS tends to equalize scores while AECS differentiates node importance. The work broadens the applicability of controllability-based centrality to PDEs and other infinite-dimensional systems, with implications for node weighting and control design in spatially distributed networks.

Abstract

Centrality analysis in dynamical network systems is essential for understanding system behavior. In finite-dimensional settings, controllability scores -- namely, the Volumetric Controllability Score (VCS) and the Average Energy Controllability Score (AECS) -- are defined as the unique solutions of specific optimization problems. In this work, we extend these concepts to infinite-dimensional systems by formulating analogous optimization problems. Moreover, we prove that these optimization problems have optimal solutions under weak assumptions, and that both VCS and AECS remain unique in the infinite-dimensional context under appropriate assumptions. The uniqueness of the controllability scores is essential to use them as a centrality measure, since it not only reflects the importance of each state in the dynamical network but also provides a consistent basis for interpretation and comparison across different researchers. Finally, we illustrate the behavior of VCS and AECS with a numerical experiment based on the heat equation.

Extension of Controllability Score to Infinite-Dimensional Systems

TL;DR

The paper tackles the problem of defining centrality in infinite-dimensional dynamical systems by extending two controllability scores, the Volumetric Controllability Score (VCS) and the Average Energy Controllability Score (AECS), through optimization over a diagonal input structure with weights summing to one. It proves the existence of optimal solutions under weak assumptions and establishes uniqueness under additional structural conditions, using the controllability Gramian in the infinite-dimensional setting and analyzing eigenvalue-based objective forms. The authors provide a detailed special-case analysis for self-adjoint generators and demonstrate the approach with heat-equation-based numerical experiments, showing how VCS tends to equalize scores while AECS differentiates node importance. The work broadens the applicability of controllability-based centrality to PDEs and other infinite-dimensional systems, with implications for node weighting and control design in spatially distributed networks.

Abstract

Centrality analysis in dynamical network systems is essential for understanding system behavior. In finite-dimensional settings, controllability scores -- namely, the Volumetric Controllability Score (VCS) and the Average Energy Controllability Score (AECS) -- are defined as the unique solutions of specific optimization problems. In this work, we extend these concepts to infinite-dimensional systems by formulating analogous optimization problems. Moreover, we prove that these optimization problems have optimal solutions under weak assumptions, and that both VCS and AECS remain unique in the infinite-dimensional context under appropriate assumptions. The uniqueness of the controllability scores is essential to use them as a centrality measure, since it not only reflects the importance of each state in the dynamical network but also provides a consistent basis for interpretation and comparison across different researchers. Finally, we illustrate the behavior of VCS and AECS with a numerical experiment based on the heat equation.

Paper Structure

This paper contains 16 sections, 14 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Contribution
  4. Outline
  5. Preliminaries
  6. Notation
  7. Controllability score for finite-dimensional systems
  8. Controllability scores for infinite-dimensional systems
  9. Optimization Problems for VCS and AECS
  10. Existence of VCS and AECS
  11. Uniqueness of VCS and AECS
  12. VCS for a special case
  13. Remark
  14. Numerical Experiments
  15. Concluding remarks
  16. ...and 1 more sections

Key Result

Lemma 2.1

\newlabellem:minimum_energy_to_fixed_point_finite_dimension0 If system eq:input_network is controllable, where $x_f$ is any point in $\mathbb{R}^n$.

Theorems & Definitions (30)

  • Lemma 2.1: zabczyk2020mathematical, Proposition 1.1.
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2
  • ...and 20 more