Resilient Structural Stabilizability of Undirected Networks

TL;DR

This paper establishes graph-theoretic conditions for the structural stabilizability of undirected networks modeled as LTI systems, and proposes algorithms for optimal actuator attack and recovery strategies.

Contribution

It provides a necessary and sufficient graph-theoretic condition for structural stabilizability and introduces approximation algorithms for actuator attack and recovery problems.

Findings

Derived a graph-theoretic condition for stabilizability.

Proposed a method to infer the maximum stabilizable subspace.

Developed a (1-1/e) approximation algorithm for recovery.

Abstract

In this paper, we consider the structural stabilizability problem of undirected networks. More specifically, we are tasked to infer the stabilizability of an undirected network from its underlying topology, where the undirected networks are modeled as continuous-time linear time-invariant (LTI) systems involving symmetric state matrices. Firstly, we derive a graph-theoretic necessary and sufficient condition for structural stabilizability of undirected networks. Then, we propose a method to infer the maximum dimension of stabilizable subspace solely based on the network structure. Based on these results, on one hand, we study the optimal actuator-disabling attack problem, i.e., removing a limited number of actuators to minimize the maximum dimension of stabilizable subspace. We show this problem is NP-hard. On the other hand, we study the optimal recovery problem with respect to the same kind of attacks, i.e., adding a limited number of new actuators such that the maximum dimension of stabilizable subspace is maximized. We prove the optimal recovery problem is also NP-hard, and we develop a (1-1/e) approximation algorithm to this problem.