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Finding Conditions for Target Controllability under Christmas Trees

Marco Peruzzo, Giacomo Baggio, Francesco Ticozzi

TL;DR

This work addresses identifying target-controllable node sets in directed networks from a structural standpoint. It introduces Christmas trees, a blended topology combining cycles and tree-like branches, and develops a graph-theoretic target-controllability condition for elementary Christmas trees by revealing a block-diagonal structure of the output controllability matrix, denoted by $\hat{\mathcal{R}}_k$, across periodic classes with diagonal weights $D^{(i)}$. The approach yields a sufficient criterion that extends beyond existing criteria (such as the $k$-walk and stem-cycle conditions) and can identify target-controllable sets not detectable by prior results, while remaining applicable to general networks through spanning subgraphs of the Christmas-tree class. The findings illuminate how simple, expressive topologies can capture nontrivial controllability properties and point toward a more complete characterization of structural target controllability in broader network classes, with future work aiming to achieve necessity and extend to full Christmas trees.

Abstract

This paper presents new graph-theoretic conditions for structural target controllability of directed networks. After reviewing existing conditions and highlighting some gaps in the literature, we introduce a new class of network systems, named Christmas trees, which generalizes trees and cacti. We then establish a graph-theoretic characterization of sets of nodes that are structurally target controllable for a simple subclass of Christmas trees. Our characterization applies to general network systems by considering spanning subgraphs of Christmas tree class and allows us to uncover target controllable sets that existing criteria fail to identify.

Finding Conditions for Target Controllability under Christmas Trees

TL;DR

This work addresses identifying target-controllable node sets in directed networks from a structural standpoint. It introduces Christmas trees, a blended topology combining cycles and tree-like branches, and develops a graph-theoretic target-controllability condition for elementary Christmas trees by revealing a block-diagonal structure of the output controllability matrix, denoted by , across periodic classes with diagonal weights . The approach yields a sufficient criterion that extends beyond existing criteria (such as the -walk and stem-cycle conditions) and can identify target-controllable sets not detectable by prior results, while remaining applicable to general networks through spanning subgraphs of the Christmas-tree class. The findings illuminate how simple, expressive topologies can capture nontrivial controllability properties and point toward a more complete characterization of structural target controllability in broader network classes, with future work aiming to achieve necessity and extend to full Christmas trees.

Abstract

This paper presents new graph-theoretic conditions for structural target controllability of directed networks. After reviewing existing conditions and highlighting some gaps in the literature, we introduce a new class of network systems, named Christmas trees, which generalizes trees and cacti. We then establish a graph-theoretic characterization of sets of nodes that are structurally target controllable for a simple subclass of Christmas trees. Our characterization applies to general network systems by considering spanning subgraphs of Christmas tree class and allows us to uncover target controllable sets that existing criteria fail to identify.

Paper Structure

This paper contains 6 sections, 5 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural target controllability and existing graph-theoretic results
  4. Christmas trees
  5. Target controllability of elementary Christmas trees
  6. Conclusions

Key Result

Proposition 1

Consider the structured system eq:sys and assume it is input accessible. A set of nodes $\mathcal{T}\subseteq \mathcal{X}$ is structurally target controllable if it can be covered by a disjoint union of stems and cycles in $\mathcal{G}$.

Figures (3)

  • Figure 1: Counterexample to the sufficiency of condition in moothedath2019target. Fig. \ref{['fig:counter_example']}(a) shows the graph describing a structured system which is not structurally target controllable with target set $\mathcal{T}=\{1,3,4,5\}$. The generic dimension of the controllability subspace is $4$. In Fig. \ref{['fig:counter_example']}(b) it is depicted the bipartite graph introduced in the aforementioned work to check structural target controllability of $\Sigma$. The upper nodes corresponds to $\mathcal{T}$, the bottom nodes are associated to path lengths $\{1,2,3,4\}$. The graph presents a perfect matching. The edges of the perfect matching are solid, the edges not considered in the matching are dashed.
  • Figure 2: Example of graph $\mathcal{G}=(\mathcal{W},\mathcal{E})$ of a Christmas tree.
  • Figure 3: Graph $\mathcal{G}$ of an elementary Christmas tree network used for comparing the conditions in Theorem \ref{['thm:target_contr']} with the existing structural target controllability conditions. White nodes belong to the set $\mathcal{X}_1 \subset \mathcal{X}$, black nodes are in the set $\mathcal{X}_0 \subset \mathcal{X}$.

Theorems & Definitions (17)

  • Definition 1: Structural output controllability
  • Definition 2: Structural target controllability
  • Proposition 1: Stem-cycle condition
  • Proposition 2: $k$-walk condition
  • Example 1: Counterexample to conjecture in moothedath2019target
  • Definition 3: Christmas tree
  • Definition 4
  • Definition 5: Elementary Christmas tree
  • Lemma 1
  • proof
  • ...and 7 more