Minimal driver sets on path and cycle graphs with arbitrary non-zero weights

TL;DR

This paper classifies minimal driver sets for path and cycle graphs with arbitrary non-zero weights, analyzing their controllability and strong structural controllability under specific matrix conditions.

Contribution

It provides a complete classification of minimal driver sets for path and cycle graphs for all sizes and identifies which sets ensure strong structural controllability.

Findings

Minimal driver sets for $P_n$ and $C_n$ are classified for all $n$.

Conditions for strong structural controllability are determined.

Laplacian matrices are excluded from the controllability analysis.

Abstract

Let $G$ be a simple, undirected graph on the vertex set $V=\{1,2,\ldots ,n\}$ and let $A$ be the adjacency matrix of $G.$ A non-empty subset $ \{i_{1},i_{2},\ldots ,i_{k}\}$ of $V$ is called a driver set for $G$ if the system $\mathbf{\dot{x}}=A\mathbf{x}+u_{1}\mathbf{e}_{i_{1}}+\cdots +u_{k} \mathbf{e}_{i_{k}}$ is controllable. In this paper we classify the minimal driver sets for the path and cycle graphs $P_{n}$ and $C_{n}$ for all values of $n$ and we determine which of those minimal driver sets render the system to be strongly structural controllable with respect to the family of all symmetric matrices $X$ satisfying $x_{ij}=0\Leftrightarrow a_{ij}=0.$ Note that this new type of strong structural controllability requires all diagonal elements of the system matrix to be equal to zero so for example the Laplacian matrix is not included in the family. Keywords: System, graph, (structural) controllability, driver set. MSC: 05C50, 05C69, 93B05, 93B25